Proof. Put h_N(x)=V(Nx)/x, with an arbitrary value at x=0. Before computing, we make the periodization limit explicit. Let \mathcal A_p be the reduced residues in \{0,\ldots,p-1\}, and for R\ge1 define the finite sum Z_{N,P}^{(R)}(\alpha)= \sum_{p\le P}\frac1p \sum_{a\in\mathcal A_p}\sum_{|j|\le R} h_N(p\alpha-a-pj). For z=p\alpha-a\in[-p,p], periodicity of V gives, when j\ge2, h_N(z-pj)+h_N(z+pj) = V(Nz)\left(\frac1{z-pj}+\frac1{z+pj}\right), and hence \sup_{|z|\le p} \left|\sum_{j>R}\bigl(h_N(z-pj)+h_N(z+pj)\bigr)\right| \ll \sum_{j>R}\frac{|z|}{p^2j^2} \ll\frac1R. The finitely many apparent quotient singularities are canceled by the zeros of the numerator and leave only bounded one-sided jumps, because V(Nx)=O_N(|x|) at an integer. Thus Z_{N,P}^{(R)} converges in L^2(\mathbb T), in fact uniformly away from those finitely many jump points, to Z_{N,P}. Compare this residue-wise cutoff |j|\le R with the literal symmetric cutoff |q|\le pR. In a fixed residue class q=a+pj, the two index intervals differ by at most two endpoint indices, and the corresponding summands are O_p(R^{-1}). Thus their difference is O_p(R^{-1}). For an arbitrary literal cutoff radius S, take R=\lfloor S/p\rfloor. The cutoffs |q|\le S and |q|\le pR differ by O_p(1) endpoints, each of size O_p(S^{-1}). Hence the residue-wise convention and the symmetric principal value in q used in (2.2) have the same limit (with p\le P fixed here).

To justify the principal-value calculation, first integrate over \eta<|x|<R, insert the absolutely and uniformly convergent series (2.1), and then let \eta\downarrow0 symmetrically and R\to\infty. The first limit can be passed through the series because, uniformly near zero, \sum_{k\ge1}\frac{|1-\cos(2\pi kNx)|}{k^2|x|}\ll N; split the sum at k=(N|x|)^{-1}. For the second limit, the symmetric truncated integrals of each exponential divided by x are bounded uniformly in its frequency and in R, by the elementary boundedness of the sine integral. Thus \sum k^{-2}<\infty supplies domination. Equivalently one may insert e^{-\epsilon|x|}, retain the symmetric principal value at zero, and let \epsilon\downarrow0; for every real a, \lim_{\epsilon\downarrow0} \operatorname{PV}\int_{\mathbb R}\frac{e(-ax)e^{-\epsilon|x|}}x\,\mathrm dx =-\mathrm i\pi\operatorname{sgn}(a), with value zero at a=0. The distributional identity \operatorname{PV}\int_{\mathbb R}\frac{e(-sx)}x\,\mathrm dx=-\mathrm i\pi\operatorname{sgn}(s) and the second series in (2.1) show that the k-th summand contributes -\mathrm i/(2\pi k^2) precisely when kN>s, zero when kN<s, and half at equality. This proves (2.3).

We may now compute with the finite periodization. The changes of variables t=p\alpha and x=t-a-pj give, for every integer n\ge0, \widehat Z_{N,P}^{(R)}(n) = \sum_{p\le P}\frac1{p^2} \sum_{a\in\mathcal A_p}e(-na/p) \int_{-a-pR}^{\,p-a+pR}h_N(x)e(-nx/p)\,\mathrm dx. Indeed, as j ranges from -R to R, the corresponding intervals of length p are adjacent and tile the displayed interval. Replacing its endpoints by \pm pR changes the integral by O_p(R^{-1}), since |h_N(x)|\ll |x|^{-1} there. Hence, for n>0, the integral tends to I_N(n/p); for n=0, it tends to zero because h_N is odd and the same endpoint estimate reduces the limit to a symmetric integral. Since \sum_{a\in\mathcal A_p}e(-na/p)=c_p(-n)=c_p(n), the L^2-limit Z_{N,P}^{(R)}\to Z_{N,P} and continuity of Fourier coefficients on L^2 prove (2.4) and \widehat Z_{N,P}(0)=0. Finally Z_{N,P} is real, which gives the negative coefficients by conjugacy. ◻

Proof. Use c_p(n)=\sum_{a\mid(p,n)}a\,\mu(p/a) and write p=ar. The left side of (2.5) becomes \sum_{a\mid n}\frac1a \sum_{\substack{r,k\ge1\\rk>n/(aN)}} \frac{\mu(r)}{r^2k^2}. On grouping by m=rk, \sum_{rk>X}\frac{\mu(r)}{r^2k^2} =\sum_{m>X}\frac1{m^2}\sum_{r\mid m}\mu(r) = \begin{cases} 1,&X<1,\\ 0,&X>1, \end{cases} with midpoint value 1/2 at X=1. Replacing a by d=n/a gives (2.5).

Since \psi(x)=\sum_{m\ne0}\frac{e(mx)}{2\pi\mathrm im} \quad\text{in }L^2(\mathbb T), the positive n-th coefficient of S_N is -\frac{\mathrm i}{2\pi n}\sum_{\substack{d\mid n\\d\le N}}d. Combining this with Lemma 2.1 and (2.5) yields exactly the coefficient of S_N-\frac12\psi(N\alpha). Negative coefficients are conjugate, and both sides have mean zero. ◻

Proof. Let q(t)=\operatorname{nint}(t). Möbius inversion and the cotangent principal-value identity give \begin{equation} \label{eq:mobcot} \operatorname{PV}\sum_{\substack{q\in\mathbb Z\\(p,q)=1}}\frac1{t-q} =\sum_{d\mid p}\mu(d)\frac{\pi}{d}\cot\left(\frac{\pi t}{d}\right). \tag{2.8} \end{equation} Put L_d(u)= \frac{\mathbf 1_{\{\left\lVert u\right\rVert_{\mathbb T}<1/(2d)\}}}{u-\operatorname{nint}(u)} -\pi\cot(\pi u), where the first quotient is interpreted in the unique local lift. The poles cancel. Since V(N(t-q))=V(Nt), writing p=dr in the difference between the nearest coprime pole and (2.8) yields \begin{equation} \label{eq:window-decomp} Y_{N,P}-Z_{N,P} =\sum_{dr\le P}\frac{\mu(d)}{d^2r}F_{N,d}(r\alpha), \qquad F_{N,d}(u)=V(Ndu)L_d(u). \tag{2.9} \end{equation}

For m\ne0, direct integration on (-1/(2d),1/(2d)) gives \begin{equation} \label{eq:Lhat} \widehat L_d(m)= \mathrm i\pi\operatorname{sgn}(m)-2\mathrm i\operatorname{Si}(\pi m/d),\qquad \left\lvert\widehat L_d(m)\right\rvert\ll\min\left(1,\frac d{|m|}\right), \tag{2.10} \end{equation} and \widehat L_d(0)=0. If v_\ell are the coefficients in (2.1), then \widehat F_{N,d}(m)=\sum_{\ell\in\mathbb Z}v_\ell \widehat L_d(m-\ell Nd). For fixed \ell, the contribution to the n-th Fourier coefficient of (2.9) is \begin{equation} \label{eq:C-ell} C_\ell(n)= \sum_{\substack{dr\le P\\r\mid n}} \frac{\mu(d)}{d^2r} \widehat L_d(n/r-\ell Nd). \tag{2.11} \end{equation}

We use the elementary estimate \begin{equation} \label{eq:Lweighted} \sum_{m\ge1}\sigma_{-1}(m) \left\lvert\widehat L_d(m-M)\right\rvert^2 \ll d\bigl(1+\log\log(e^e+|M|+d)\bigr). \tag{2.12} \end{equation} Indeed, on |m-M|\le d, the first bound in (2.10) and \sigma_{-1}(m)\ll1+\log\log(e^e+m) give \sum_{\substack{m\ge1\\|m-M|\le d}} \sigma_{-1}(m)|\widehat L_d(m-M)|^2 \ll d\bigl(1+\log\log(e^e+|M|+d)\bigr). On the annulus 2^jd<|m-M|\le2^{j+1}d, the second bound in (2.10) gives \sum_{\substack{m\ge1\\2^jd<|m-M|\le2^{j+1}d}} \sigma_{-1}(m)|\widehat L_d(m-M)|^2 \ll 2^{-j}d \bigl(1+\log\log(e^e+|M|+2^{j+1}d)\bigr). Here \log\log(e^e+|M|+2^{j+1}d) \ll \log\log(e^e+|M|+d)+\log(j+2), and both \sum_{j\ge0}2^{-j} and \sum_{j\ge0}2^{-j}\log(j+2) converge. Summing the annuli proves (2.12).

Apply Cauchy–Schwarz to (2.11) with the fixed weights d^{-3/2}r^{-1}. Since \sum_{\substack{dr\le P\\r\mid n}}d^{-3/2}r^{-1} \ll\sigma_{-1}(n), this gives |C_\ell(n)|^2 \ll \sigma_{-1}(n) \sum_{\substack{dr\le P\\r\mid n}} d^{-5/2}r^{-1} \left|\widehat L_d(n/r-\ell Nd)\right|^2. Sum over n=rm, use \sigma_{-1}(rm)\le\sigma_{-1}(r)\sigma_{-1}(m), and then (2.12) with M=\ell Nd. We obtain \sum_{n\ge1}|C_\ell(n)|^2 \ll \log(2P)\bigl(1+\log\log(e^e+|\ell|NP)\bigr), because \sum_{r\le R}\sigma_{-1}(r)/r\ll\log(2R) and \sum_d d^{-3/2}<\infty. The negative frequencies are conjugate, while the zero coefficient vanishes because V(Ndu) is even and L_d(u) is odd. Finally, \sum_{\ell\in\mathbb Z}|v_\ell| \sqrt{1+\log\log(e^e+|\ell|NP)} \ll \sqrt{1+\log\log(e^e+NP)} by |v_\ell|\ll(1+\ell^2)^{-1}. Minkowski and Plancherel prove (2.7). ◻

Proof. Expanding c_p(n), the positive n-th coefficient, apart from the fixed factor -\mathrm i/(2\pi), is A_P(n)= \sum_{d\mid n}\frac1d \sum_{r>P/d}\frac{\mu(r)}{r^2} H_*\left(\frac{n}{drN}\right). Since H_*(u)\ll\min(1,u^{-1}), dropping the Möbius signs gives |A_P(n)|\ll \begin{cases} \tau(n)/P,&n\le NP,\\[2mm] \displaystyle \frac{N\tau(n)}n\left(1+\log\frac{n}{NP}\right),&n>NP. \end{cases} The first line follows by summing r^{-2} above P/d. For the second, split the r-sum at n/(dN). Now use \sum_{n\le X}\tau(n)^2\ll X(1+\log X)^3 to obtain \sum_{n\le NP}|A_P(n)|^2 \ll \frac1{P^2}\sum_{n\le NP}\tau(n)^2 \ll \frac NP\bigl(1+\log(NP)\bigr)^3. For 2^jNP<n\le2^{j+1}NP, j\ge0, the second pointwise bound similarly gives \sum_{2^jNP<n\le2^{j+1}NP}|A_P(n)|^2 \ll \frac NP\,2^{-j}(j+2)^2 \bigl(1+\log(2^{j+1}NP)\bigr)^3. The last expression is summable in j, with total O((N/P)(1+\log(NP))^3). Summing the dyadic blocks proves (2.13). ◻

Proof. Insert c_p(n)=\sum_{d\mid(p,n)}d\,\mu(p/d) twice. The number of multiples of \operatorname{lcm}(d,e) in I is |I|/\operatorname{lcm}(d,e)+O(1). The main term vanishes when p\ne p', by complete-period orthogonality of Ramanujan sums. Indeed, averaging their exponential definitions over one period \operatorname{lcm}(p,p') leaves only pairs of reduced fractions satisfying a/p+b/p'\in\mathbb Z; reduction forces p=p'. The absolute sum of the endpoint errors is at most the right side of (2.14). ◻

Proof. For integers u\le v, put A_u(k)=\sum_{u\le n\le k}a_n. Ordinary summation by parts gives the exact identity \begin{equation} \label{eq:discrete-Abel-identity} \sum_{n=u}^v a_nW(n) =A_u(v)W(v)+ \sum_{n=u}^{v-1}A_u(n)\bigl(W(n)-W(n+1)\bigr). \tag{2.17} \end{equation} By (2.15), its absolute value is at most M\left(|W(v)|+ \sum_{n=u}^{v-1}|W(n+1)-W(n)|\right). On either infinite tail, both the endpoint value and the remaining variation tend to zero. Thus the finite sums satisfy the Cauchy criterion. Letting the endpoints tend to the ends of the index set in (2.17) proves (2.16); the displayed factor 2 harmlessly covers all one-sided and two-sided endpoint conventions. ◻

Proof. Put L=\log N and P=N(1+L)^{10}. For a dyadic block Q<p\le2Q, Q>N, let A_Q(n)=\sum_{Q<p\le2Q}\frac{c_p(-n)}{p^2} H_*\left(\frac{n}{pN}\right),\qquad n>0. Write h_p(n)=H_*(n/(pN)). Expanding the square gives \begin{equation} \label{eq:natural-block-expansion} \sum_{n\ge1}|A_Q(n)|^2 =\sum_{p,p'\asymp Q}\frac1{p^2p'^2} \sum_{n\ge1}c_p(-n)c_{p'}(-n)h_p(n)h_{p'}(n). \tag{2.20} \end{equation} The midpoint convention in H_* preserves monotonicity. More precisely, at every integer m\ge1, H_*(m^-)-H_*(m)=H_*(m)-H_*(m^+)=\frac1{2m^2}. Consequently each h_p is nonnegative and nonincreasing, tends to zero, and is bounded by \zeta(2). Hence h_ph_{p'} is also nonnegative and nonincreasing, and \left\lVert h_ph_{p'}\right\rVert_\infty+ \sum_{n\ge1}|h_p(n+1)h_{p'}(n+1)-h_p(n)h_{p'}(n)| \le2\zeta(2)^2. For p\ne p', apply Lemma 2.6 to a_n=c_p(-n)c_{p'}(-n), and use Lemma 2.5. Thus the inner sum in (2.20) is O(\sigma_1(p)\sigma_1(p')), including every midpoint jump and every Abel boundary term. The off-diagonal contribution is therefore \sum_{\substack{p,p'\asymp Q\\p\ne p'}} \frac{\sigma_1(p)\sigma_1(p')}{p^2p'^2} \ll(\log\log(3Q))^2. For the diagonal, use \sum_{n\bmod p}c_p(n)^2=p\varphi(p),\qquad \sum_{n\ge1}H_*(n/(pN))^2\ll pN. More explicitly, on the j-th period jp<n\le(j+1)p, monotonicity of H_* and complete-period orthogonality give a contribution at most p\varphi(p)H_*(j/N)^2. Since H_*(u)\ll\min(1,u^{-1}), \sum_{j\ge0}H_*(j/N)^2\ll N. Thus subdivision into periods of length p gives \sum_{p\asymp Q}\frac1{p^4} \sum_{n\ge1}c_p(n)^2H_*(n/(pN))^2 \ll\sum_{p\asymp Q}\frac{N\varphi(p)}{p^3} \ll\frac NQ. Thus \begin{equation} \label{eq:high-block} \left\lVert A_Q\right\rVert_{\ell^2(n\ge1)}^2 \ll1+(\log\log(3Q))^2. \tag{2.21} \end{equation} There are O(\log(P/N))=O(\log L) blocks. Minkowski, (2.21), and Lemma 2.4 give \left\lVert Z_{N,\infty}-Z_{N,N}\right\rVert_2 \ll(\log L)^2+o(1)=o(L), which is (2.18).

Finally combine (2.6), Proposition 2.3 with P=N, and (2.18). The term \frac12\psi(N\alpha) has bounded L^2-norm, while \left\lVert Y_{N,N}-Z_{N,N}\right\rVert_2 \ll\sqrt{L\log L}=o(L). This proves (2.19). ◻

Proof. Put C_x(n)=\sum_{q\le x}c_q(n). Set x_{\rm CK}=x,\qquad y_{\rm CK}=2K,\qquad B_{\rm CK}=11. When x\le\sqrt {y_{\rm CK}}, their elementary formula \sum_{n\le y_{\rm CK}}|C_x(n)|^2 =\frac{y_{\rm CK}x^2}{2\zeta(2)} +O(x^4+xy_{\rm CK}\log x) is O(Kx^2). When \sqrt {y_{\rm CK}}<x\le T:=K/(\log K)^{B_{\rm tail}}, apply part (ii) of their theorem with the displayed substitution. We verify both of its range hypotheses, including the lower logarithmic margin. Since B_{\rm CK}=11, B_{\rm tail}>30, and x\le T, x(\log x)^{2B_{\rm CK}} \le T(\log T)^{22} \le K(\log K)^{22-B_{\rm tail}} \le 2K=y_{\rm CK} for all sufficiently large K. Since x>\sqrt {y_{\rm CK}}, we also have y_{\rm CK}<x^2\le x^2(\log x)^{B_{\rm CK}}. The bounded remaining values of K are absorbed in the constant. Moreover, \log(xy_{\rm CK})\asymp\log x\asymp\log K throughout this range. The function denoted by \kappa in that theorem satisfies \sup_{u\in\mathbb R}|\kappa(u)|<\infty, so its main term is O(y_{\rm CK}x^2)=O(Kx^2). Its error relative to y_{\rm CK}x^2 is O\!\left((\log x)^{10} \left(x^{-1/2}+(y_{\rm CK}/x)^{-1/2}\right)\right)=O(1) when B_{\rm tail}>30: indeed x^{-1/2}\le y_{\rm CK}^{-1/4}, and y_{\rm CK}/x\ge2(\log K)^{B_{\rm tail}} at the upper endpoint of the range. Thus \begin{equation} \label{eq:CK} \sum_{n\le2K}|C_x(n)|^2\ll Kx^2 \qquad (2\le x\le K/(\log K)^{B_{\rm tail}}). \tag{3.2} \end{equation} Partial summation gives \sum_{Q<q\le R}\frac{c_q(n)}{q^2} = \frac{C_R(n)}{R^2}-\frac{C_Q(n)}{Q^2} +2\int_Q^R\frac{C_x(n)}{x^3}\,\mathrm dx. Minkowski and (3.2) bound the part below T=K/(\log K)^{B_{\rm tail}} by C\sqrt K/Q. Above T, the divisor formula gives |C_x(n)|\le x\tau(n),\qquad \left\lVert C_x\right\rVert_{\ell^2(n\le2K)} \ll x\sqrt K(\log K)^{3/2}. Its contribution is \ll\frac{\sqrt K(\log K)^{3/2}}T =\frac{(\log K)^{B_{\rm tail}+3/2}}{\sqrt K}, and hence the argument has proved the more precise intermediate estimate \begin{equation} \label{eq:ram-tail-two-term} \left\|\sum_{q>Q}\frac{c_q(n)}{q^2}\right\|_{\ell^2(n\le2K)} \ll \frac{\sqrt K}{Q} +E_K,\qquad E_K:=\frac{(\log K)^{B_{\rm tail}+3/2}}{\sqrt K}, \quad 2\le Q\le T. \tag{3.3} \end{equation} The same estimate holds for finite tails, and the same divisor calculation also gives, uniformly in the upper endpoint, \begin{equation} \label{eq:ram-tail-terminal} \left\|\sum_{x<q\le R}\frac{c_q(n)}{q^2}\right\|_{\ell^2(n\le2K)} \ll\frac{\sqrt K(\log K)^{3/2}}{x}, \qquad x\ge T. \tag{3.4} \end{equation} For all sufficiently large K, E_K=o(1), whereas \sqrt K/Q\ge1 in the stated range; the bounded remaining values are absorbed into the constant. Letting R\to\infty proves the infinite tail. Notice that the larger range Q\le K/(\log K)^{B_{\rm tail}} would not follow from this argument: at its upper endpoint the displayed terminal error is too large by a factor (\log K)^{3/2}. The range stated here is exactly the one used below. ◻

Proof. Set z_\varepsilon(v)=\beta(|v|/\varepsilon),\qquad h(u)=1-\eta(|u|),\qquad h_a(v)=h(v/a). Because a\le\varepsilon/4, the support of h_a is contained in \{|v|\le2a\}, where z_\varepsilon=1. On that support, z_\varepsilon-h_a=\eta(|v|/a)=\rho_{a,\varepsilon}; off that support, h_a=0 and \eta(|v|/a)=1. Hence \begin{equation} \label{eq:rho-two-profiles} \rho_{a,\varepsilon}=z_\varepsilon-h_a. \tag{3.10} \end{equation} Both terms on the right are fixed smooth compactly supported profiles, the second followed by dilation by a.

For every fixed integer M\ge2, integration by parts M times and scaling give, for t\ge0, \begin{equation} \label{eq:rho-Fourier-envelope} \sup_{1\le r\le2}|\widehat\rho_{a,\varepsilon}(rt)| \le C_{\varepsilon,M}\left\{ \varepsilon(1+\varepsilon t)^{-M} +a(1+at)^{-M}\right\}. \tag{3.11} \end{equation} Since J_a'(t)=-2\pi\mathrm i\,\widehat\rho_{a,\varepsilon}(t), the integral of the right side of (3.11) over t>0 is O_{\varepsilon,M}(1). This proves (3.8) with the requested explicit envelope.

We next prove (3.7). The function \rho_{a,\varepsilon} is even, so W_a is odd and J_a(t)=-2\mathrm i\int_0^\infty \rho_{a,\varepsilon}(v)\frac{\sin(2\pi tv)}v\,\mathrm dv. Since \frac{\sin(2\pi tv)}v\,\mathrm dv=\,\mathrm d\operatorname{Si}(2\pi tv), Stieltjes integration by parts yields |J_a(t)| \le2\sup_{x\in\mathbb R}|\operatorname{Si}(x)| \operatorname{TV}_{(0,\infty)}(\rho_{a,\varepsilon}) \ll_\varepsilon1. The total variation is uniform in a, either directly from the two transition regions or from (3.10). Similarly, Fourier integration by parts gives |tJ_a'(t)| =2\pi|t\widehat\rho_{a,\varepsilon}(t)| \le\left\lVert\rho_{a,\varepsilon}'\right\rVert_1 \ll_\varepsilon1. At the origin, J_a(0)=0 by oddness and |J_a'(t)|\le2\pi\left\lVert\rho_{a,\varepsilon}\right\rVert_1 \ll_\varepsilon1. Thus |J_a(t)|+|tJ_a'(t)|\ll_\varepsilon |t| \qquad (|t|\le1).

It remains to record the high-frequency estimates with their dependence on the order. Leibniz’s rule, the Gevrey bounds for the two transition profiles, and their supports imply, for m\ge1, \begin{equation} \label{eq:W-L1-derivatives} \left\lVert W_a^{(m)}\right\rVert_1 \le C_\varepsilon^{m+1}(m!)^2a^{-m}, \qquad \left\lVert\rho_{a,\varepsilon}^{(m+1)}\right\rVert_1 \le C_\varepsilon^{m+1}(m!)^2a^{-m}. \tag{3.12} \end{equation} For the first estimate, expand W_a^{(m)} =\sum_{k=0}^m \binom mk\rho_{a,\varepsilon}^{(k)} \left(\frac1v\right)^{(m-k)}. For k=0, integration over a<|v|<\varepsilon gives O(m!a^{-m}). For k\ge1, the derivative of \rho is supported in a<|v|<2a,\qquad \varepsilon/2<|v|<\varepsilon. The inner region contributes at most C_\varepsilon^{m+1}(m!)^2 a^{-k}a^{-(m-k+1)}a \ll C_\varepsilon^{m+1}(m!)^2a^{-m}, and the outer region is smaller. The second estimate in (3.12) follows in the same way by scaling. Integrating by parts m times in J_a, and m+1 times in \widehat\rho, now gives |J_a(t)|+|tJ_a'(t)| \le C_\varepsilon^{m+1}(m!)^2(a|t|)^{-m}. Together with the small- and intermediate-frequency bounds, this proves (3.7).

The same Leibniz expansion in L^2 gives \left\| \rho_{a,\varepsilon}^{(k)} \left(\frac1v\right)^{(j-k)} \right\|_2 \le C_\varepsilon^{j+1}(j!)^2a^{-j-1/2}. For k=0, integrate |v|^{-2(j+1)} from a to \varepsilon. For k\ge1, the inner transition region contributes a^{-k}a^{-(j-k+1)}a^{1/2}=a^{-j-1/2}, and the outer region is smaller. Summing over k proves (3.9). ◻

Proof. For fixed A,\varepsilon, the relation a=A/L\le\varepsilon/4 holds for all sufficiently large N, which is the range relevant to the limsup. Put W=W_a, J=J_a, and b_p(\alpha)= \mathbf 1_{\{(p,q_p)=1\}}\frac{\rho(p\delta_p)}{p\delta_p}. For each p, there is the exact locally finite periodization \begin{equation} \label{eq:bp-cell-periodization} b_p(\alpha)= \sum_{\substack{q\in\mathbb Z\\(p,q)=1}} W\left(p^2\left(\alpha-\frac qp\right)\right). \tag{3.14} \end{equation} Indeed, the support of the summand centered at q/p has radius \varepsilon/p^2. Distinct centers are separated by 1/p, and 2\varepsilon/p^2<1/p. Thus the supports are disjoint. On such a support, |p\alpha-q|<\varepsilon/p<1/2, so q=q_p, and p^2(\alpha-q/p)=p\delta_p. Since W\in C_c^\infty(\mathbb R) and vanishes with all derivatives at its cutoff boundaries, (3.14) also proves that b_p is a periodic C^\infty function. Restricting to reduced q’s removes entire disjoint smooth cells and creates no jump.

Changing variables in those cells gives, for every n\in\mathbb Z, \begin{equation} \label{eq:near-bhat} \begin{aligned} \widehat b_p(n) &=\frac1{p^2} \sum_{\substack{q\bmod p\\(p,q)=1}}e(-nq/p) \int_{\mathbb R}W(v)e(-nv/p^2)\,\mathrm dv\\ &=\frac{c_p(-n)}{p^2}J(n/p^2). \end{aligned} \tag{3.15} \end{equation} In particular J(0)=0, since W is odd.

We begin with the constant coefficient v_0=1/12 in (2.1). Here and below in this proof, n\asymp K means K<n\le2K. Put Q_K=\sqrt{aK},\qquad R_K=\sqrt{\varepsilon K}. Lemma 3.2 and p\frac{\,\mathrm d}{\,\mathrm dp}J(n/p^2) =-2\frac n{p^2}J'(n/p^2) give, for every fixed integer m\ge1, \begin{equation} \label{eq:Jbounds} |J(n/p^2)|+ p\left|\frac{\,\mathrm d}{\,\mathrm dp}J(n/p^2)\right| \ll_{m,\varepsilon} \begin{cases} (p/Q_K)^{2m},&p<Q_K,\\ 1,&Q_K\le p\le R_K,\\ K/p^2,&p>R_K. \end{cases} \tag{3.16} \end{equation} For the last line, use the small-t part of (3.7) when n/p^2\le1. If 1<n/p^2\le2/\varepsilon, the uniform bound is at most C_\varepsilon K/p^2: indeed p^2<n\le2K, so K/p^2\ge1/2, and the uniform O_\varepsilon(1) bound is absorbed by the displayed quantity. We now write every boundary term in the multiplier summation. Let \mathcal H_K=\ell^2\{n\in\mathbb Z:K<n\le2K\},\qquad u_p=\left(\frac{c_p(-n)}{p^2}\right)_{n\asymp K}, \qquad w_x=\bigl(J(n/x^2)\bigr)_{n\asymp K}. For 2\le U<\infty, define the finite tail \mathcal R_{x,U}=\sum_{x<p\le U}u_p. Writing \odot for coordinatewise multiplication, summation by parts in each coordinate gives the exact finite-dimensional identity \begin{equation} \label{eq:near-vector-Abel} \sum_{2<p\le U}u_p\odot w_p = \mathcal R_{2,U}\odot w_2+ \int_2^U\mathcal R_{x,U}\odot \frac{\,\mathrm dw_x}{\,\mathrm dx}\,\mathrm dx. \tag{3.17} \end{equation} Indeed, x\mapsto\mathcal R_{x,U} has jump -u_p at p and \mathcal R_{U,U}=0, so Stieltjes integration by parts is -\int_{(2,U]}w_x\odot\,\mathrm d\mathcal R_{x,U} =\mathcal R_{2,U}\odot w_2+ \int_2^U\mathcal R_{x,U}\odot\,\mathrm dw_x. This proves (3.17), including both endpoint terms.

Let T=\frac K{(\log K)^{B_{\rm tail}}},\qquad E_K=\frac{(\log K)^{B_{\rm tail}+3/2}}{\sqrt K}. Since Ramanujan sums are real and even, the finite-tail form of (3.3) gives, uniformly for 2\le x\le U\le T, \begin{equation} \label{eq:near-finite-tail} \left\lVert\mathcal R_{x,U}\right\rVert_{\mathcal H_K} \ll\frac{\sqrt K}{x}+E_K. \tag{3.18} \end{equation} For a fixed m\ge1, put \omega_{K,m}(x)= \begin{cases} (x/Q_K)^{2m},&x<Q_K,\\ 1,&x\ge Q_K. \end{cases} Equation (3.16) yields \begin{equation} \label{eq:near-weight-envelope} \left\lVert w_x\right\rVert_{\ell^\infty(n\asymp K)} +x\left\lVert\frac{\,\mathrm dw_x}{\,\mathrm dx}\right\rVert_{\ell^\infty(n\asymp K)} \ll_{\varepsilon,m}\omega_{K,m}(x). \tag{3.19} \end{equation} The uniform envelope in Lemma 3.2 also gives \begin{equation} \label{eq:J-total-variation} \int_0^\infty \left\lVert\frac{\,\mathrm dw_x}{\,\mathrm dx}\right\rVert_{\ell^\infty(n\asymp K)} \,\mathrm dx\ll_\varepsilon1. \tag{3.20} \end{equation} To check the change of variables, write n=rK, 1\le r\le2, and t=K/x^2; the left side is at most a constant times \int_0^\infty\sup_{1\le r\le2}r|J'(rt)|\,\mathrm dt, which is (3.8).

Assume first K\ge16/a, so Q_K\ge4, and set U_0=\min\{N,T\}. Equations (3.17)–(3.20) give \begin{align} \left\|\sum_{2<p\le U_0}u_p\odot w_p\right\|_{\mathcal H_K} &\le \left(\frac{\sqrt K}{2}+E_K\right)\left\lVert w_2\right\rVert_\infty +\int_2^{U_0} \left(\frac{\sqrt K}{x}+E_K\right) \left\lVert\frac{\,\mathrm dw_x}{\,\mathrm dx}\right\rVert_\infty\,\mathrm dx \notag\\ &\ll_{\varepsilon,m} \sqrt K\left\{ \frac{\omega_{K,m}(2)}2+ \int_2^\infty\frac{\omega_{K,m}(x)}{x^2}\,\mathrm dx \right\}+E_K \notag\\ &\ll_{\varepsilon,m}\frac{\sqrt K}{Q_K}+E_K. \label{eq:near-Abel-low}\tag{3.21}\end{align} The last line follows from Q_K^{-2m}\int_2^{Q_K}x^{2m-2}\,\mathrm dx +\int_{Q_K}^\infty x^{-2}\,\mathrm dx\ll_m Q_K^{-1}, \qquad \omega_{K,m}(2)\ll_m Q_K^{-1}. If T<N, apply (3.17) on (T,N]. The terminal tail estimate (3.4) and (3.19) give \begin{align} \left\|\sum_{T<p\le N}u_p\odot w_p\right\|_{\mathcal H_K} &\ll_\varepsilon \sqrt K(\log K)^{3/2} \left(\frac1T+\int_T^\infty\frac{\,\mathrm dx}{x^2}\right)\notag\\ &\ll_\varepsilon \frac{\sqrt K(\log K)^{3/2}}T=E_K. \label{eq:near-Abel-terminal}\tag{3.22}\end{align} The finitely many terms p\le2 satisfy the same O_{\varepsilon,m}(\sqrt K/Q_K) bound by the first line of (3.16). Consequently, whenever K\ge16/a, uniformly over the dyadic block, \begin{equation} \label{eq:near-constant-block} \left\| \sum_{p\le N}\frac{c_p(n)}{p^2}J(n/p^2) \right\|_{\ell^2(n\asymp K)} \ll_\varepsilon \frac{\sqrt K}{Q_K}+E_K \ll_\varepsilon a^{-1/2}. \tag{3.23} \end{equation} The assertion about E_K is uniform, not merely pointwise. Indeed, for all sufficiently large N, the function x\mapsto(\log x)^{B_{\rm tail}+3/2}x^{-1/2} is decreasing throughout x\ge16/a, and therefore \begin{equation} \label{eq:near-E-uniform} \sup_{K\ge16/a}E_K \le C\sqrt a\,\bigl(\log(16/a)\bigr)^{B_{\rm tail}+3/2}=o(1). \tag{3.24} \end{equation}

We spell out both ends of the frequency sum. For 1\le n\le16/a, absolute convergence, the uniform bound for J, and the divisor formula give \begin{equation} \label{eq:near-initial-pointwise} \left|\sum_{p\le N}\frac{c_p(-n)}{p^2}J(n/p^2)\right| \ll\sum_{p\ge1}\frac{|c_p(n)|}{p^2} \ll\sigma_{-1}(n). \tag{3.25} \end{equation} Consequently, using \sigma_{-1}(n)\ll1+\log\log(3n), these frequencies contribute at most \begin{equation} \label{eq:near-initial-sum} O_\varepsilon\left( a^{-1}(1+\log\log(1/a))^2\right)=o(L/a) \tag{3.26} \end{equation} to the squared norm.

At the other endpoint suppose that Q_K>N, equivalently K>N^2/a. For all sufficiently large N, this implies N<T, because x/(\log x)^{B_{\rm tail}} is then increasing for x\ge N^2/a, and T\ge\frac{N^2/a}{(\log(N^2/a))^{B_{\rm tail}}}>N. Apply (3.17) with U=N. Every x\le N then lies below Q_K, and hence \begin{equation} \label{eq:near-low-envelope-tail} \left\lVert w_x\right\rVert_\infty+x\left\lVert w_x'\right\rVert_\infty \ll_{\varepsilon,m}(x/Q_K)^{2m}. \tag{3.27} \end{equation} Using (3.18) in the exact Abel identity gives \begin{align} \left\|\sum_{2<p\le N}u_p\odot w_p\right\|_{\mathcal H_K} &\ll_{\varepsilon,m} \sqrt K Q_K^{-2m} \left(1+\int_2^N x^{2m-2}\,\mathrm dx\right) +E_K\left(\frac N{Q_K}\right)^{2m}\notag\\ &\ll_{\varepsilon,m} \frac{\sqrt K}{Q_K} \left(\frac N{Q_K}\right)^{2m-1} +E_K\left(\frac N{Q_K}\right)^{2m}. \label{eq:near-terminal-block-detail}\tag{3.28}\end{align} The terms p\le2 obey the same first bound directly. By (3.24), E_K\le a^{-1/2}=\sqrt K/Q_K for all large N. Since N/Q_K<1, we conclude that \begin{equation} \label{eq:near-terminal-block} \left\|\sum_{p\le N}\frac{c_p(-n)}{p^2}J(n/p^2) \right\|_{\ell^2(n\asymp K)} \ll_{\varepsilon,m} a^{-1/2}\left(\frac N{Q_K}\right)^{2m-1}. \tag{3.29} \end{equation} On the dyadic blocks K_r=2^rN^2/a, the square of the last bound is at most \frac1a\left(\frac{N^2}{aK_r}\right)^{2m-1} =\frac1a\cdot2^{-r(2m-1)}. Thus the complete squared tail K>N^2/a is O_{\varepsilon,m}(a^{-1}). Between 16/a and N^2/a there are O(L) dyadic blocks, and each contributes O_\varepsilon(a^{-1}) by (3.23). Together with (3.26), this proves that the constant mode has squared norm \begin{equation} \label{eq:near-constant} \ll_\varepsilon L/a=\frac{L^2}{A}. \tag{3.30} \end{equation}

Negative frequencies contribute the same amount by conjugation, and the zero frequency vanishes because J(0)=0.

Now fix \ell\ne0. Since N,\ell,q_p are integers, e(\ell N\delta_p)=e(\ell Np\alpha). Differentiating the disjoint-cell representation (3.14) and changing variables in every cell gives, for every integer j\ge0, \begin{align} \left\lVert b_p^{(j)}\right\rVert_2^2 &=\varphi(p)p^{4j-2}\left\lVert W^{(j)}\right\rVert_2^2,\notag\\ \left\lVert b_p^{(j)}\right\rVert_2 &\le C_\varepsilon^{j+1}(j!)^2 p^{2j-1/2}a^{-j-1/2}\notag\\ &=C_\varepsilon^{j+1}(j!)^2 \left(\frac{p^2}{a}\right)^j(ap)^{-1/2}. \label{eq:bp-derivative}\tag{3.31}\end{align} In particular, \begin{equation} \label{eq:bp-L2} \left\lVert b_p\right\rVert_2^2\ll_\varepsilon\frac1{ap}. \tag{3.32} \end{equation} Whenever b_p(\alpha)\ne0, \left|\alpha-\frac{q_p}p\right| =\frac{|p\delta_p|}{p^2}<\frac1{2p^2}, and (p,q_p)=1. Legendre’s criterion therefore makes p a continued-fraction denominator of \alpha. Since such denominators satisfy Q_{n+2}\ge2Q_n, only an absolute bounded number of active p’s can lie in a dyadic interval. Let P_0=N\exp(-L^{1/3}). The same recurrence shows that the total number of active denominators in (P_0,N] is O(L^{1/3}). Pointwise Cauchy–Schwarz and (3.32) therefore give \begin{align} \left\|\sum_{P_0<p\le N}b_p(\alpha)e(\ell Np\alpha)\right\|_2^2 &\ll L^{1/3}\sum_{P_0<p\le N}\left\lVert b_p\right\rVert_2^2\notag\\ &\ll\frac{L^{1/3}}a\sum_{P_0<p\le N}\frac1p \ll\frac{L^{2/3}}a=o(L/a). \label{eq:near-highp}\tag{3.33}\end{align}

For p\le P_0, divide into dyadic blocks \mathcal P_s=(P_s/2,P_s], and set F_{s,\ell}(\alpha)= \sum_{p\in\mathcal P_s}b_p(\alpha)e(\ell Np\alpha). The same pointwise sparsity and (3.32) give \begin{equation} \label{eq:near-Qblock} \left\lVert F_{s,\ell}\right\rVert_2^2 \ll\sum_{p\in\mathcal P_s}\left\lVert b_p\right\rVert_2^2 \ll_\varepsilon\frac1a. \tag{3.34} \end{equation} Let \Pi_{s,\ell} be Fourier projection onto the signed annulus \mathcal A_{s,\ell}= \left\{n\in\mathbb Z:\ \operatorname{sgn}(n)=\operatorname{sgn}(\ell),\quad \frac{|\ell|NP_s}{4}\le|n|\le2|\ell|NP_s\right\}. We use the following exact leakage inequality. If f\in H^j(\mathbb T), m_0\in\mathbb Z, and every integer outside a frequency set \mathcal A has distance at least D from m_0, then Plancherel gives \begin{align} \left\lVert(I-\Pi_{\mathcal A})(e(m_0\alpha)f)\right\rVert_2^2 &=\sum_{\substack{k\in\mathbb Z\\m_0+k\notin\mathcal A}} |\widehat f(k)|^2\notag\\ &\le D^{-2j}\sum_{k\in\mathbb Z}|k|^{2j}|\widehat f(k)|^2 =(2\pi D)^{-2j}\left\lVert f^{(j)}\right\rVert_2^2. \label{eq:near-Fourier-leakage}\tag{3.35}\end{align} For p\in(P_s/2,P_s], the carrier m_0=\ell Np lies in \mathcal A_{s,\ell}, and its distance from the complement is at least D_{s,\ell}=|\ell|NP_s/4. Applying (3.35), the triangle inequality, and (3.31), we obtain \begin{align} \left\lVert(I-\Pi_{s,\ell})F_{s,\ell}\right\rVert_2 &\le(2\pi D_{s,\ell})^{-j} \sum_{p\in\mathcal P_s}\left\lVert b_p^{(j)}\right\rVert_2\notag\\ &\le C_\varepsilon^{j+1}(j!)^2|\ell|^{-j} \left(\frac{P_s}{Na}\right)^j \left(\frac{P_s}{a}\right)^{1/2}. \label{eq:near-leakage}\tag{3.36}\end{align} There is no unrecorded dependence on j in this estimate. Choose j=\left\lceil 2L^{2/3}\right\rceil. Since \log\frac{P_s}{Na}\le-L^{1/3}+\log(L/A), |\ell|^{-j}\le1, and \log(j!)\le j\log j, the logarithm of the last line of (3.36) is at most -jL^{1/3}+j\log(L/A)+2j\log j+O_\varepsilon(j) +\frac12\log(P_s/a) =-\frac32L+O_\varepsilon(L^{2/3}\log L). It follows that \begin{equation} \label{eq:near-leakage-small} \sup_{s,\,\ell\ne0} \left\lVert(I-\Pi_{s,\ell})F_{s,\ell}\right\rVert_2=o(N^{-1}). \tag{3.37} \end{equation}

The annuli \mathcal A_{s,\ell}, with P_s dyadic, have overlap multiplicity at most an absolute constant B. Consequently, frequency by frequency and then by Plancherel, \begin{align} \left\|\sum_s\Pi_{s,\ell}F_{s,\ell}\right\|_2^2 &=\sum_{n\in\mathbb Z} \left|\sum_{s:n\in\mathcal A_{s,\ell}} \widehat F_{s,\ell}(n)\right|^2\notag\\ &\le B\sum_s\sum_{n\in\mathcal A_{s,\ell}} |\widehat F_{s,\ell}(n)|^2 \le B\sum_s\left\lVert F_{s,\ell}\right\rVert_2^2 \ll_\varepsilon L/a. \label{eq:near-carrier-square}\tag{3.38}\end{align} There are O(L) blocks, so (3.37) also gives \sum_s\left\lVert(I-\Pi_{s,\ell})F_{s,\ell}\right\rVert_2=o(1). The triangle inequality and (3.38) therefore give \begin{equation} \label{eq:near-lowp-shifted} \left\|\sum_{p\le P_0}b_p(\alpha)e(\ell Np\alpha)\right\|_2^2 \ll_\varepsilon L/a, \tag{3.39} \end{equation} uniformly for \ell\ne0. Together with (3.33), this is uniform in \ell\ne0. Finally, the absolutely convergent Fourier series V(x)=\sum_{\ell\in\mathbb Z}v_\ell e(\ell x) gives Y^{\mathrm{near}}_{N,A,\varepsilon} =\sum_{\ell\in\mathbb Z}v_\ell \sum_{p\le N}b_p(\alpha)e(\ell N\delta_p). Use (3.30) for \ell=0, the shifted estimate for \ell\ne0, Minkowski’s inequality, and \sum_{\ell\ne0}|v_\ell|<\infty. We obtain \left\lVert Y^{\mathrm{near}}_{N,A,\varepsilon}\right\rVert_2^2 \ll_\varepsilon\frac La=\frac{L^2}{A}, which proves (3.13). ◻

Proof. Fix temporarily 0<\delta<\varepsilon/2, and take an even smooth function \chi=\chi_{\varepsilon,\delta} such that \chi(v)=0\quad(|v|\le\varepsilon-\delta),\qquad \chi(v)=1\quad(|v|\ge\varepsilon). Let Y_N^\chi be (1.1) with the extra factor \chi(p\delta_p), and define its full periodization by Z_N^\chi(\alpha)= \sum_{p\le N}\frac1p\, \operatorname{PV}\sum_{\substack{q\in\mathbb Z\\(p,q)=1}} \frac{V(N(p\alpha-q))}{p\alpha-q}\, \chi\bigl(p(p\alpha-q)\bigr). Because 1-\chi is supported in |p(p\alpha-q)|<\varepsilon<1/2, there is at most one q in its support, necessarily q_p. Therefore \begin{equation} \label{eq:far-window} Y_N^\chi-Z_N^\chi=Y_N-Z_{N,N}, \tag{3.42} \end{equation} and Proposition 2.3 gives \begin{equation} \label{eq:far-window-L2} \left\lVert Y_N^\chi-Z_N^\chi\right\rVert_2^2\ll L\log L=o(L^2). \tag{3.43} \end{equation}

Set \begin{equation} \label{eq:Rchi} R_\chi(t)=\operatorname{PV}\int_{\mathbb R}\frac{\chi(v)}v e(-tv)\,\mathrm dv. \tag{3.44} \end{equation} Put \kappa_\chi=1-\chi. This function is even and compactly supported. For t\ne0, the Fourier transform of \operatorname{PV}(1/v) gives \begin{equation} \label{eq:Rchi-decomposition} R_\chi(t)=-\mathrm i\pi\operatorname{sgn}(t)- \operatorname{PV}\int_{\mathbb R}\frac{\kappa_\chi(v)}v e(-tv)\,\mathrm dv. \tag{3.45} \end{equation} The second term in (3.45) tends to zero as t\to0. Indeed, after pairing v and -v, its absolute value is at most 2\int_0^C\kappa_\chi(v) \frac{|\sin(2\pi tv)|}{v}\,\mathrm dv=O_\chi(|t|). Thus the symmetric value and the two one-sided limits are \begin{equation} \label{eq:Rchi-jump} R_\chi(0)=0,\qquad R_\chi(0^+)=-\mathrm i\pi,qquad R_\chi(0^-)=\mathrm i\pi. \tag{3.46} \end{equation} For t\ne0, differentiation in (3.45) gives R_\chi'(t)=2\pi\mathrm i\widehat{\kappa_\chi}(t). Dirichlet’s integral applied to (3.44) shows that R_\chi(t)\to0 at both infinities. Since \widehat{\kappa_\chi} is Schwartz, integration from the appropriate infinity shows that R_\chi is smooth and rapidly decreasing on each open half-line. More precisely, for all r,J\ge0, \begin{equation} \label{eq:Rchi-Schwartz} |R_\chi^{(r)}(t)| \le C_{r,J,\chi}(1+|t|)^{-J}\qquad(t\ne0), \tag{3.47} \end{equation} where for r=0 the two half-lines are treated separately. In particular, R_\chi, with the value in (3.46), is a regulated BV function with one jump; it is not being regarded as a Schwartz function on all of \mathbb R.

Auxiliary scaled-multiplier estimate. Let R=R_\chi. If p,p'\in(Q,2Q] and a,a'\in\mathbb R, then, for every J\ge0, \begin{equation} \label{eq:scaled-BV-general} \begin{split} W(t)&=R\left(\frac{t-a}{p^2}\right) \overline{R\left(\frac{t-a'}{p'^2}\right)},\\ \left\lVert W\right\rVert_\infty+\operatorname{TV}_{\mathbb R}(W) &\ll_{J,\chi} \left(1+\frac{|a-a'|}{Q^2}\right)^{-J}. \end{split} \tag{3.48} \end{equation} Here the values at a and a' are induced by R(0)=0. Moreover, suppose that \ell\ne0, a=\ell Np, a'=\ell Np', and put H=\frac{|\ell|N}{Q},\qquad I_{\ell,Q}=\left\{t:\ \operatorname{sgn}(t)=\operatorname{sgn}(\ell),\quad \frac{|\ell|NQ}{2}\le|t|\le3|\ell|NQ\right\}, \qquad E_{\ell,Q}=\mathbb R\setminus I_{\ell,Q}. Then \begin{equation} \label{eq:scaled-BV-projected} \left\lVert\mathbf 1_{E_{\ell,Q}}W\right\rVert_\infty+ \operatorname{TV}_{\mathbb R}(\mathbf 1_{E_{\ell,Q}}W) \ll_{J,\chi}H^{-J}. \tag{3.49} \end{equation} The convention at the two endpoints of I_{\ell,Q} is immaterial.

Proof of the auxiliary estimate. Write E_M(x)=(1+|x|)^{-M}. Uniformly for 1/4\le\lambda\le4, d\in\mathbb R, and M>J+1, \begin{equation} \label{eq:two-envelope-convolution} \sup_x E_M(x)E_M(\lambda x-d)+ \int_{\mathbb R}E_M(x)E_M(\lambda x-d)\,\mathrm dx \ll_{J,M}(1+|d|)^{-J}. \tag{3.50} \end{equation} To verify this, split the line into |x|\ge|d|/8 and its complement. On the first set one extracts (1+|d|)^{-J} from the first factor and integrates the remaining envelope. On the complement, |\lambda x-d|\ge|d|/2, and one instead extracts the same power from the second factor. The supremum estimate is identical.

Away from a,a', ordinary differentiation gives W'(t)=\frac1{p^2}R'\left(\frac{t-a}{p^2}\right) \overline{R\left(\frac{t-a'}{p'^2}\right)} +\frac1{p'^2}R\left(\frac{t-a}{p^2}\right) \overline{R'\left(\frac{t-a'}{p'^2}\right)}. In the first integral use t=a+p^2x, and in the second use t=a'+p'^2x. Since p^2/p'^2\in[1/4,4], (3.47) and (3.50) bound both \int_{\mathbb R\setminus\{a,a'\}}|W'(t)|\,\mathrm dt and \left\lVert W\right\rVert_\infty by the right side of (3.48).

It remains to count the point jumps. If a\ne a', then the total variation contributed at a, including the assigned value W(a)=0, is 2\pi\left|R\left(\frac{a-a'}{p'^2}\right)\right|; the analogous contribution at a' is 2\pi|R((a'-a)/p^2)|. Both have the required separation decay. If a=a', the two one-sided products are bounded and the assigned value is zero, so the total point contribution is at most 2\pi^2. This proves (3.48), including unequal scales and all possible coincidences of the jump points.

For (3.49), both centers lie strictly inside I_{\ell,Q}, while for every t\in E_{\ell,Q}, 1+\frac{|t-a|}{p^2}\gg1+H, \qquad 1+\frac{|t-a'|}{p'^2}\gg1+H. Thus multiplication by \mathbf 1_{E_{\ell,Q}} kills the original jump points. On E_{\ell,Q}, use the displayed derivative formula, take the supremum of one far-away factor, and integrate the other. This contributes O_{J,\chi}(H^{-J}). The only new jumps are at the two endpoints of I_{\ell,Q}; both factors there have normalized distance \gg H, so these point contributions obey the same bound. This proves (3.49). \square

We also record the interchange that leads to the Fourier coefficients below. The original symmetric principal value agrees with its Abel regularization. Indeed, once |q| is large, the cutoff factor is one, and pairing q with -q gives V(Np\alpha) \left(\frac1{p\alpha-q}+\frac1{p\alpha+q}\right) =O_p(q^{-2}). Thus the paired tails are absolutely summable, and Abel’s theorem applies to the resulting convergent symmetric series. To compare the literal Abel weights with a symmetrically weighted pairing, note that, for q>|p\alpha|, the two weights are e^{-\lambda_{\rm A}q}e^{\pm\lambda_{\rm A}p\alpha}. Replacing both by e^{-\lambda_{\rm A}q} changes the paired tail by O_p\!\left( \lambda_{\rm A}\sum_{q\ge1}\frac{e^{-\lambda_{\rm A}q}}q \right) =O_p\!\left(\lambda_{\rm A}\log(2/\lambda_{\rm A})\right)=o(1). The symmetrically weighted paired series is dominated by its O_p(q^{-2}) tail and therefore converges to the original symmetric principal value. We may therefore first truncate each q-sum symmetrically and insert the Abel factor \exp(-\lambda_{\rm A}|p\alpha-q|), \lambda_{\rm A}>0. The sum is then finite and, after the truncation is removed, absolutely integrable. The Fourier series (2.1) is absolutely and uniformly convergent, so it may be inserted term by term before unfolding the residue classes modulo p. For the \ell-th term, the changes of variables x=p\alpha-q and then v=px give the damped transform \int_{\mathbb R}\frac{\chi(v)}v e\left(-\frac{n-\ell Np}{p^2}v\right) e^{-\lambda_{\rm A}|v|/p}\,\mathrm dv. Letting \lambda_{\rm A}\downarrow0 symmetrically gives R_\chi((n-\ell Np)/p^2). This passage is pointwise in the real frequency and uniformly bounded; uniform convergence across frequency zero is neither true nor needed. More precisely, for the principal kernel, \operatorname{PV}\int_{\mathbb R}\frac{e(-sv)e^{-\lambda_{\rm A}|v|/p}}v\,\mathrm dv =-2\mathrm i\arctan\left(\frac{2\pi ps}{\lambda_{\rm A}}\right), which is uniformly bounded and converges pointwise to -\mathrm i\pi\operatorname{sgn}(s). For the correction, pairing v and -v gives -2\mathrm i\int_0^C\kappa_\chi(v) \frac{\sin(2\pi sv)}v e^{-\lambda_{\rm A}v/p}\,\mathrm dv, which is uniformly bounded by Dirichlet’s estimate and converges for each s as \lambda_{\rm A}\downarrow0. Since p\le N is a finite sum and \sum_\ell|v_\ell|<\infty, dominated convergence now removes the Abel factor and exchanges the principal value, the Fourier series, and the cell unfolding. This is the same symmetric truncation convention as in Lemma 2.1.

Using (2.1), the n-th Fourier coefficient of the \ell-th mark mode of Z_N^\chi, without the factor v_\ell, is \begin{equation} \label{eq:B-ell} B_\ell(n)= \sum_{p\le N}\frac{c_p(-n)}{p^2} R_\chi\left(\frac{n-\ell Np}{p^2}\right). \tag{3.51} \end{equation}

The unshifted mode. For n\asymp K, the transition in (3.51) with \ell=0 is Q_K=\sqrt K. From (3.47), for a suitable fixed C_\chi\ge1 and every j, |R_\chi(n/p^2)|+ p\left|\frac{\,\mathrm d}{\,\mathrm dp}R_\chi(n/p^2)\right| \ll_{j,\chi} \begin{cases} (p/Q_K)^j,&p<Q_K/C_\chi,\\ 1,&p\ge Q_K/C_\chi. \end{cases} We now spell out the vector-valued Abel summation, including its boundaries. Put \mathcal H_K=\ell^2\{n\in\mathbb Z:K<n\le2K\},\qquad u_p=\left(\frac{c_p(-n)}{p^2}\right)_{n\asymp K},\qquad w_x=\bigl(R_\chi(n/x^2)\bigr)_{n\asymp K}, and, for a finite upper endpoint U, set \mathcal R_{x,U}=\sum_{x<p\le U}u_p. The exact identity (3.17), applied with this multiplier, reads \begin{equation} \label{eq:Rchi-vector-Abel} \sum_{2<p\le U}u_p\odot w_p =\mathcal R_{2,U}\odot w_2+ \int_2^U\mathcal R_{x,U}\odot\frac{\,\mathrm dw_x}{\,\mathrm dx}\,\mathrm dx. \tag{3.52} \end{equation} There is no omitted upper boundary, because \mathcal R_{U,U}=0.

For all sufficiently large K, put T=K/(\log K)^{B_{\rm tail}}\ge2 and U_0=\min\{N,T\}; the bounded set of smaller blocks will be absorbed into the final constant. The finite-tail form of (3.3) gives, uniformly for 2\le x\le U_0, \begin{equation} \label{eq:Rchi-low-finite-tail} \left\lVert\mathcal R_{x,U_0}\right\rVert_{\mathcal H_K} \ll\frac{\sqrt K}{x}+E_K. \tag{3.53} \end{equation} For a fixed integer j>2, define \omega_{K,j}(x)= \min\left\{1,\left(\frac{C_\chi x}{Q_K}\right)^j\right\}. The preceding multiplier bound implies \begin{equation} \label{eq:Rchi-weight-envelope} \left\lVert w_x\right\rVert_{\ell^\infty(n\asymp K)}+ x\left\lVert\frac{\,\mathrm dw_x}{\,\mathrm dx}\right\rVert_{\ell^\infty(n\asymp K)} \ll_{j,\chi}\omega_{K,j}(x). \tag{3.54} \end{equation} The half-line Schwartz bounds also imply the vector-valued total variation estimate \begin{equation} \label{eq:Rchi-vector-variation} \int_0^\infty\sup_{K\le n\le2K} \left|\frac{\,\mathrm d}{\,\mathrm dx}R_\chi(n/x^2)\right|\,\mathrm dx\ll_\chi1. \tag{3.55} \end{equation} To see this, put n=rK, 1\le r\le2, and t=K/x^2; the left side is bounded by a constant times \int_0^\infty\sup_{1\le r\le2}r|R_\chi'(rt)|\,\mathrm dt. This is finite by (3.47). Applying (3.52), (3.53), and (3.54), and using \frac{\omega_{K,j}(2)}2+ \int_2^\infty\frac{\omega_{K,j}(x)}{x^2}\,\mathrm dx \ll_{j,\chi}\frac1{Q_K}, we obtain \begin{equation} \label{eq:Rchi-low-Abel-bound} \left\|\sum_{2<p\le U_0}u_p\odot w_p\right\|_{\mathcal H_K} \ll_{j,\chi}\frac{\sqrt K}{Q_K}+E_K =O_\chi(1)+o(1). \tag{3.56} \end{equation} The E_K-part includes the boundary at 2 and is bounded using (3.55).

If T<N, apply the same exact identity to (T,N]. The finite form of (3.4) gives \left\lVert\mathcal R_{x,N}\right\rVert_{\mathcal H_K} \ll\frac{\sqrt K(\log K)^{3/2}}x, \qquad T\le x\le N, where now \mathcal R_{x,N}=\sum_{x<p\le N}u_p. Moreover, \left|\frac{\,\mathrm d}{\,\mathrm dx}R_\chi(n/x^2)\right| \ll_\chi\min(x^{-1},Kx^{-3}). The lower boundary at T is O(\sqrt K(\log K)^{3/2}/T)=O(E_K), since R_\chi is bounded; the upper boundary again vanishes. The remaining Abel integral is \ll \sqrt K(\log K)^{3/2}K\int_T^\infty x^{-4}\,\mathrm dx =o(E_K). The finitely many terms p\le2 satisfy an O_\chi(1) bound by the first multiplier estimate. We have therefore proved \begin{equation} \label{eq:B0-block} \left\lVert B_0\right\rVert_{\ell^2(n\asymp K)}\ll_\chi1. \tag{3.57} \end{equation} We also record the high-frequency tail rather than subsume it in (3.57). Suppose that Q_K=\sqrt K>N, equivalently K>N^2. For all sufficiently large N, this implies T>N: the function x/(\log x)^{B_{\rm tail}} is increasing in this range, and at x=N^2 it is larger than N. Furthermore, once N is sufficiently large, the function x\mapsto(\log x)^{B_{\rm tail}+3/2}x^{-1/2} is decreasing on [N^2,\infty). Hence \sup_{K\ge N^2}E_K \le \frac{(\log N^2)^{B_{\rm tail}+3/2}}N=o(1), so every use of E_K=o(1) below is uniform over the high-frequency blocks. Apply (3.52) with U=N. Every x\le N lies below Q_K, so (3.54) and (3.53) give, for each fixed j>2, \begin{align} \left\|\sum_{2<p\le N}u_p\odot w_p\right\|_{\mathcal H_K} &\ll_{j,\chi} \sqrt K Q_K^{-j} \left(1+\int_2^Nx^{j-2}\,\mathrm dx\right) +E_K\left(\frac N{Q_K}\right)^j\notag\\ &\ll_{j,\chi} \left(\frac N{Q_K}\right)^{j-1}. \label{eq:Rchi-high-frequency-block}\tag{3.58}\end{align} In the last step we used \sqrt K/Q_K=1, E_K=o(1), and N/Q_K<1. The terms p\le2 obey the same bound directly from (3.47), after increasing the fixed decay exponent if necessary. On the dyadic blocks K_r=2^rN^2, the square of (3.58) is O_{j,\chi}(2^{-r(j-1)}), and is therefore summable. The range K\le N^2 contains O(\log N)=O(L) dyadic blocks, each bounded by (3.57).

Finally, the reality of \chi(v)/v gives R_\chi(-t)=\overline{R_\chi(t)}, and Ramanujan sums are real and even. Hence B_0(-n)=\overline{B_0(n)}, so the negative-frequency blocks have the same bounds. At frequency zero, B_0(0)=0 by (3.46). Combining the positive, negative, and zero frequencies proves \begin{equation} \label{eq:B0-total} \sum_n|B_0(n)|^2\ll_\chi L. \tag{3.59} \end{equation}

Shifted modes. Fix \ell\ne0. For p,p'\asymp Q, apply the auxiliary scaled-multiplier estimate above with a=\ell Np and a'=\ell Np'. It gives, for W_{p,p'}(t)= R_\chi\left(\frac{t-\ell Np}{p^2}\right) \overline{R_\chi\left(\frac{t-\ell Np'}{p'^2}\right)}, \begin{equation} \label{eq:BV-product} \left\lVert W_{p,p'}\right\rVert_\infty+\operatorname{TV}_{\mathbb R}(W_{p,p'}) \ll_{J,\chi} \left(1+\frac{|\ell|N|p-p'|}{Q^2}\right)^{-J}. \tag{3.60} \end{equation} Before making the dyadic decomposition, remove the singleton p=1. Since c_1(n)=1, its contribution is B_{\ell}^{[1]}(n)=R_\chi(n-\ell N). Translation by the integer \ell N, together with (3.47) and R_\chi(0)=0, gives the uniform bound \begin{equation} \label{eq:shifted-p-one} \sum_{n\in\mathbb Z}|B_{\ell}^{[1]}(n)|^2 =\sum_{m\in\mathbb Z}|R_\chi(m)|^2\ll_\chi1. \tag{3.61} \end{equation} Thus it remains to decompose the terms 2\le p\le N into the blocks Q<p\le2Q below; the singleton can be restored at the end by the triangle inequality. The function W_{p,p'} tends to zero at both infinities. Therefore Lemmas 2.6 and 2.5, together with (3.60), imply, when p\ne p', \begin{equation} \label{eq:shifted-cross} \left|\sum_{n\in\mathbb Z}c_p(n)c_{p'}(n)W_{p,p'}(n)\right| \ll_{J,\chi}\sigma_1(p)\sigma_1(p') \left(1+\frac{|\ell|N|p-p'|}{Q^2}\right)^{-J}. \tag{3.62} \end{equation} Let B_{\ell,Q} denote the dyadic Q<p\le2Q part of (3.51). We first write out its diagonal contribution. Put a=\ell Np. Since a/p=\ell N\in\mathbb Z, every interval (a+kp,a+(k+1)p]\cap\mathbb Z is a complete residue system modulo p, and hence \sum_{a+kp<n\le a+(k+1)p}c_p(n)^2=p\varphi(p). The half-line bounds (3.47) now give \begin{align} \sum_{n\in\mathbb Z}c_p(n)^2 \left|R_\chi\left(\frac{n-\ell Np}{p^2}\right)\right|^2 &\le p\varphi(p)\sum_{k\in\mathbb Z} \sup_{k/p<x\le(k+1)/p}|R_\chi(x)|^2\notag\\ &\ll_\chi p^2\varphi(p). \label{eq:shifted-diagonal-periods}\tag{3.63}\end{align} Consequently the diagonal part of \left\lVert B_{\ell,Q}\right\rVert_2^2 is \ll_\chi\sum_{Q<p\le2Q}\frac{\varphi(p)}{p^2}\ll_\chi1.

For the off-diagonal, put \lambda=|\ell|N/Q^2, use \sigma_1(p)/p\ll\log\log(3Q), and choose J>2. Equation (3.62) gives \begin{align} &\sum_{\substack{p,p'\asymp Q\\p\ne p'}} \frac{\sigma_1(p)\sigma_1(p')}{p^2p'^2} (1+\lambda|p-p'|)^{-J}\notag\\ &\qquad\ll \frac{(\log\log(3Q))^2}{Q^2} Q\sum_{1\le h\le Q}(1+\lambda h)^{-J}\notag\\ &\qquad\ll (\log\log(3Q))^2 \min\left(1,\frac{Q}{|\ell|N}\right). \label{eq:shifted-offdiagonal-sum}\tag{3.64}\end{align} Combining this with (3.63) proves, uniformly for \ell\ne0, the slightly stronger form \begin{equation} \label{eq:shifted-Q} \left\lVert B_{\ell,Q}\right\rVert_2^2 \ll_\chi 1+(\log\log(3Q))^2 \min\left(1,\frac{Q}{|\ell|N}\right). \tag{3.65} \end{equation}

It remains to square-sum in Q. Let Q_0=\frac{N}{L^{C_{\rm far}}},\qquad C_{\rm far}>10, and let \Pi_{\ell,Q} project onto \operatorname{sgn}(n)=\operatorname{sgn}(\ell),\qquad \frac{|\ell|NQ}{2}\le|n|\le3|\ell|NQ. Put H=|\ell|N/Q and let E_{\ell,Q} be the complement in \mathbb R of the displayed carrier annulus. Expanding by Plancherel gives the exact identity \begin{equation} \label{eq:carrier-expansion} \begin{split} \left\lVert(1-\Pi_{\ell,Q})B_{\ell,Q}\right\rVert_2^2 ={}&\sum_{p,p'\asymp Q}\frac1{p^2p'^2} \sum_{n\in\mathbb Z}c_p(n)c_{p'}(n)\\ &\hspace{18mm}\cdot \mathbf 1_{E_{\ell,Q}}(n)W_{p,p'}(n). \end{split} \tag{3.66} \end{equation} For p\ne p', the projected estimate (3.49) and Lemmas 2.6 and 2.5 bound the inner sum by \begin{equation} \label{eq:carrier-offdiagonal-inner} O_{J,\chi}\bigl(\sigma_1(p)\sigma_1(p')H^{-J}\bigr). \tag{3.67} \end{equation}

For p=p', every retained n satisfies |n-\ell Np|/p^2\gg H. Subdivide into the periods used in (3.63) and take a Schwartz exponent larger than J+1. The remaining supremum sum is bounded by p\int_{|x|\gg H}(1+|x|)^{-J-2}\,\mathrm dx+O(H^{-J-2}) \ll_JpH^{-J}. It follows that \begin{equation} \label{eq:carrier-diagonal-inner} \sum_{n\in E_{\ell,Q}\cap\mathbb Z}c_p(n)^2 \left|R_\chi\left(\frac{n-\ell Np}{p^2}\right)\right|^2 \ll_{J,\chi}p^2\varphi(p)H^{-J}. \tag{3.68} \end{equation} After applying the outside weights in (3.66), the diagonal contribution is O_{J,\chi}(H^{-J}), because \sum_{p\asymp Q}\varphi(p)/p^2\ll1. The off-diagonal contribution from (3.67) is O_{J,\chi}((\log\log(3Q))^2H^{-J}). We have therefore proved, including both projection-boundary jumps, that for every J, \begin{equation} \label{eq:carrier-leak} \left\lVert(1-\Pi_{\ell,Q})B_{\ell,Q}\right\rVert_2^2 \ll_{J,\chi} (1+\log\log(3Q))^2 \left(\frac{Q}{|\ell|N}\right)^J. \tag{3.69} \end{equation}

The carrier annuli have bounded overlap as Q ranges dyadically: for each integer frequency n, at most a fixed number of dyadic values of Q can satisfy n\in I_{\ell,Q}, since those values have ratio at most 6. Therefore \begin{equation} \label{eq:carrier-bounded-overlap} \begin{split} \left\|\sum_Q\Pi_{\ell,Q}B_{\ell,Q}\right\|_2^2 &=\sum_n\left|\sum_Q \mathbf 1_{I_{\ell,Q}}(n)B_{\ell,Q}(n)\right|^2\\ &\ll\sum_Q\left\lVert B_{\ell,Q}\right\rVert_2^2. \end{split} \tag{3.70} \end{equation} For Q\le Q_0, the sum of (3.65) over dyadic Q is O_\chi(L): the constant terms contribute O(L), while \sum_{Q\le Q_0}(\log\log(3Q))^2\frac{Q}{|\ell|N}=o(1). Moreover, after taking square roots in (3.69), \sum_{Q\le Q_0} \left\lVert(1-\Pi_{\ell,Q})B_{\ell,Q}\right\rVert_2=o(1) on choosing J large. Bounded overlap of the projected annuli in (3.70) therefore gives \begin{equation} \label{eq:low-Q} \left\|\sum_{Q\le Q_0}B_{\ell,Q}\right\|_2^2\ll_\chi L. \tag{3.71} \end{equation} There are O(\log L) blocks above Q_0. Each has squared norm O_\chi((\log L)^2), so the triangle inequality gives \sup_{\ell\ne0}\sum_n|B_\ell(n)|^2 \ll_\chi L+(\log L)^4=o(L^2). Minkowski, \sum_{\ell\ne0}|v_\ell|<\infty, and (3.59) prove \left\lVert Z_N^\chi\right\rVert_2^2\ll_\chi L+(\log L)^4=o(L^2). Together with (3.43), the same holds for Y_N^\chi.

Finally remove the smoothing. The difference between Y_N^\chi and the sharp sum in (3.40) is supported on \varepsilon-\delta<|p\delta_p|<\varepsilon. Every active p is a convergent denominator, so at most O(L) are active at one \alpha, since Q_n\ge F_{n+1} for the Fibonacci numbers F_n. Each summand is O_\varepsilon(1), and \int_0^1\sum_{p\le N} \mathbf 1_{\{\varepsilon-\delta<|p\delta_p|<\varepsilon,\, (p,q_p)=1\}}\,\mathrm d\alpha \le2\delta\sum_{p\le N}\frac{\varphi(p)}{p^2} \ll\delta L. Pointwise Cauchy–Schwarz thus bounds the squared L^2-norm of the difference by O_\varepsilon(\delta L^2). First let N\to\infty and then \delta\downarrow0. This proves (3.40).

It remains to prove the quantified uniformity (3.41). Fix 0<\delta<\varepsilon_0/2. For every t\in[\varepsilon_0,\varepsilon_1], construct \chi_{t,\delta} from one fixed smooth transition profile so that it vanishes on |v|\le t-\delta and equals one on |v|\ge t. All these profiles vanish on a fixed neighborhood of zero, and their transition points remain in a fixed compact set. Hence, for each fixed r,J, \begin{equation} \label{eq:uniform-chi-seminorms} \sup_{t\in[\varepsilon_0,\varepsilon_1]} C_{r,J,\chi_{t,\delta}}<\infty, \tag{3.72} \end{equation} where the bound may depend on the presently fixed \delta. Every periodization and multiplier estimate above is consequently uniform in t. In particular, (3.43) and the bound for Z_N^\chi give the explicit common estimate \sup_{t\in[\varepsilon_0,\varepsilon_1]} \left\lVert Y_N^{\chi_{t,\delta}}\right\rVert_2^2 \ll L\log L+C_\delta\bigl(L+(\log L)^4\bigr)=o_\delta(L^2). Equivalently, \begin{equation} \label{eq:uniform-smoothed-far} \sup_{t\in[\varepsilon_0,\varepsilon_1]} \left\lVert Y_N^{\chi_{t,\delta}}\right\rVert_2=o_\delta(L). \tag{3.73} \end{equation}

Let F_N(t) denote the sharp sum in (3.41), and put D_{N,t}=F_N(t)-Y_N^{\chi_{t,\delta}}. This difference is supported on t-\delta<|p\delta_p|<t. Each active summand is O_{\varepsilon_0}(1). Moreover, its reduced fraction satisfies \left|\alpha-\frac{q_p}{p}\right| <\frac{\varepsilon_1}{p^2}<\frac1{2p^2}. Legendre’s criterion therefore makes q_p/p a continued-fraction convergent. Since consecutive convergent denominators satisfy Q_{n+2}\ge2Q_n, the number of active p\le N is O(L), uniformly in \alpha and t. On the other hand, direct cell integration gives, uniformly in t, \begin{equation} \label{eq:uniform-transition-first-moment} \int_0^1\sum_{p\le N} \mathbf 1_{\{t-\delta<|p\delta_p|<t,\ (p,q_p)=1\}}\,\mathrm d\alpha \le2\delta\sum_{p\le N}\frac{\varphi(p)}{p^2}+O(\delta) \ll\delta L. \tag{3.74} \end{equation} The O(\delta) term harmlessly covers the endpoint cells for p=1. Pointwise Cauchy–Schwarz, followed by (3.74), yields \begin{equation} \label{eq:uniform-transition-L2} \sup_{t\in[\varepsilon_0,\varepsilon_1]} \left\lVert D_{N,t}\right\rVert_2^2\ll_{\varepsilon_0}\delta L^2. \tag{3.75} \end{equation} Combining (3.73) and (3.75), \limsup_{N\to\infty} \sup_{t\in[\varepsilon_0,\varepsilon_1]} L^{-1}\left\lVert F_N(t)\right\rVert_2 \ll_{\varepsilon_0}\sqrt\delta. Letting \delta\downarrow0 proves (3.41). Thus the order of limits is: first N\to\infty with \delta fixed, and then \delta\downarrow0. ◻

Proof. Fix 0<\varepsilon<1/4, and abbreviate v_p=p\delta_p,\qquad \xi_p=Lv_p,\qquad u_p=N\delta_p\bmod1. Let F_N(t)= \sum_{\substack{p\le N,\ (p,q_p)=1\\|v_p|\ge t}} \frac{V(u_p)}{v_p}. Proposition 3.4 gives \|F_N(\varepsilon/2)\|_2=o(L).

For all sufficiently large N, the fixed-profile cutoff used in Proposition 3.3 yields the exact decomposition, outside null boundary sets, \begin{equation} \label{eq:minor-decomposition} \sum_{\substack{p\le N,\ (p,q_p)=1\\|\xi_p|>A}} \frac{V(u_p)}{v_p} = F_N(\varepsilon/2) +Y^{\mathrm{near}}_{N,A,\varepsilon} +E^{\mathrm{lo}}_{N,A}-E^{\mathrm{up}}_{N,\varepsilon}, \tag{3.77} \end{equation} where \begin{align*} E^{\mathrm{lo}}_{N,A} &= \sum_{\substack{p\le N,\ (p,q_p)=1\\A<|\xi_p|<2A}} \left(1-\eta(|\xi_p|/A)\right)\frac{V(u_p)}{v_p},\\ E^{\mathrm{up}}_{N,\varepsilon} &= \sum_{\substack{p\le N,\ (p,q_p)=1\\ \varepsilon/2<|v_p|<\varepsilon}} \beta(|v_p|/\varepsilon)\frac{V(u_p)}{v_p}. \end{align*} This identity, rather than a containment argument, is needed because cancellation does not compare a weighted transition sum with its containing sharp band.

The near term satisfies \limsup_{N\to\infty} L^{-2}\|Y^{\mathrm{near}}_{N,A,\varepsilon}\|_2^2 \ll_\varepsilon A^{-1} by Proposition 3.3. For the lower transition, define \mathcal P to be the limiting Poisson process in Lemma 4.1, and put h_A(\xi,u)= \left(1-\eta(|\xi|/A)\right) \mathbf 1_{\{A<|\xi|<2A\}}\frac{V(u)}{\xi}. Lemma 4.1 and the continuous mapping theorem (the boundary has zero limiting intensity) give \frac{E^{\mathrm{lo}}_{N,A}}L \Longrightarrow \int h_A\,\,\mathrm d\mathcal P. The process in Lemma 4.1 uses p<N; the omitted p=N term is nonzero on a set of measure O_A((LN)^{-1}), so it does not affect this convergence. The limiting shot has mean zero by symmetry in \xi, and \operatorname{Var}\!\left(\int h_A\,\,\mathrm d\mathcal P\right) =\frac1{\zeta(2)}\int h_A(\xi,u)^2\,\mathrm d\xi\,\mathrm du \ll\frac1A. Thus Portmanteau and Chebyshev bound its contribution to (3.76) by O(A^{-1}\gamma^{-2}).

It remains to justify the upper transition. Put r_0=\varepsilon/2, r_1=\varepsilon, and h(t)=\beta(t/\varepsilon). Stieltjes layer cake gives exactly E^{\mathrm{up}}_{N,\varepsilon} =h(r_0)F_N(r_0)+\int_{r_0}^{r_1}h'(t)F_N(t)\,\mathrm dt. This identity follows first pointwise in \alpha, by applying the one-variable Stieltjes identity to each of the finitely many shots and then summing. As a map from [r_0,r_1] to the separable space L^2(\mathbb T), t\mapsto F_N(t) is strongly measurable: its joint representative is measurable in (t,\alpha). Moreover, for fixed N, throughout this interval |F_N(t,\alpha)|\le \frac{N\|V\|_\infty}{r_0}, so it is Bochner integrable. The pointwise identity is therefore also an identity in L^2(\mathbb T). Minkowski’s inequality and (3.41) therefore give \frac{\left\lVert E^{\mathrm{up}}_{N,\varepsilon}\right\rVert_2}{L} \le \left(|h(r_0)|+\int_{r_0}^{r_1}|h'(t)|\,\mathrm dt\right) \sup_{r_0\le t\le r_1}\frac{\left\lVert F_N(t)\right\rVert_2}{L} \longrightarrow0. Apply Chebyshev to the near term, use the preceding bounds in (3.77), and finally let A\to\infty. This proves (3.76). ◻

Proof. Roadmap. We first establish the unmarked factorial measures in deterministic continued-fraction index time and transfer them from Gauss to Lebesgue measure. We then prove asymptotic Haar marking by an oscillatory cylinder estimate, treating early and late configurations separately. Finally, factorial measures are upgraded to point-process convergence, index time is replaced by denominator time, and the endpoint time layers are restored by a direct rational-cell estimate.

Convergents and rare digits. Write \alpha=[0;a_1,a_2,\ldots] and denote its regular convergents by P_n/Q_n, with P_{-1}=1,\quad P_0=0,\qquad Q_{-1}=0,\quad Q_0=1. Let T(x)=\{1/x\}, x_n=T^n\alpha, and y_n=Q_{n-1}/Q_n. The determinant identities for convergents give \begin{equation} \label{eq:cf-error} Q_n\alpha-P_n=(-1)^n\frac{\theta_n}{Q_n},\qquad \theta_n=\frac{x_n}{1+x_ny_n} =\frac1{a_{n+1}+x_{n+1}+y_n}. \tag{4.5} \end{equation} These identities include n=0, with y_0=0.

Fix B<\infty. If a point in (4.3) satisfies |Lp\delta_p|\le B, then, for all sufficiently large N, \left|\alpha-\frac{q_p}{p}\right| <\frac1{2p^2}. Since q_p/p is reduced, Legendre’s criterion [9, Chapter II, §7, Theorem 19, p. 30] says that it is a convergent. Thus, on every fixed \xi-window, (4.1) is exactly \begin{equation} \label{eq:cf-process} \sum_{Q_n<N} \delta_{\left(\log Q_n/L,\;(-1)^nL\theta_n,\; (-1)^nN\theta_n/Q_n\bmod1\right)}. \tag{4.6} \end{equation}

Let \nu_G be Gauss measure, d\nu_G(x)=\frac{dx}{\log2\,(1+x)}. Set \kappa=\frac{12\log2}{\pi^2}. The continued-fraction digits are exponentially \psi-mixing: there are C>0 and \vartheta>1 such that, if F\in\sigma(a_1,\ldots,a_j),\qquad H\in\sigma(a_{j+d},a_{j+d+1},\ldots), then \begin{equation} \label{eq:gauss-mixing} |\nu_G(F\cap H)-\nu_G(F)\nu_G(H)| \le C\vartheta^{-d}\nu_G(F)\nu_G(H). \tag{4.7} \end{equation} We use the form and notation recorded in [5, (1.7)]. We record two consequences of (4.7) that will be used repeatedly. Suppose that \mathscr A and \mathscr H are \sigma-fields for which \left|\nu_G(A\cap B)-\nu_G(A)\nu_G(B)\right| \le \eta\,\nu_G(A)\nu_G(B) \qquad(A\in\mathscr A,\ B\in\mathscr H). Then, for arbitrary complex-valued U_1\in L^1(\mathscr A) and U_2\in L^1(\mathscr H), \begin{equation} \label{eq:functional-psi-mixing} \left|\mathbb E_{\nu_G}(U_1U_2) -(\mathbb E_{\nu_G}U_1)(\mathbb E_{\nu_G}U_2)\right| \le 2\eta\,\mathbb E_{\nu_G}|U_1|\,\mathbb E_{\nu_G}|U_2|. \tag{4.8} \end{equation} Indeed, for nonnegative simple functions this follows by expanding both functions into indicator functions and applying the event inequality term by term. Monotone convergence gives the result for arbitrary nonnegative integrable functions. For a complex function U_1, write U_1=(\Re U_1)^+-(\Re U_1)^- +\mathrm i\bigl((\Im U_1)^+-(\Im U_1)^-\bigr), and similarly for U_2. The sums of the L^1-norms of the four nonnegative parts are at most \sqrt2\,\mathbb E|U_1| and \sqrt2\,\mathbb E|U_2|, respectively, which proves (4.8).

A second consequence is the following quasi-Bernoulli bound. If m_1<\cdots<m_s and E_j\in\sigma(a_{m_j}), then \begin{equation} \label{eq:digit-quasi-bernoulli} \nu_G\left(\bigcap_{j=1}^sE_j\right) \le C_s\prod_{j=1}^s\nu_G(E_j). \tag{4.9} \end{equation} To see this, expose the events from left to right and apply (4.7) at each step. Every gap is at least one, so each application costs at most 1+C\vartheta^{-1}. This explains precisely what is meant below by the quasi-Bernoulli bound.

For a compact interval J\Subset(0,\infty), the last formula in (4.5) gives, on the event L\theta_n\in J, \begin{equation} \label{eq:digit-replacement} \left|L\theta_n-\frac{L}{a_{n+1}}\right| \ll_J\frac1L. \tag{4.10} \end{equation} The same conclusion, with a constant depending on a fixed enlargement of J, holds when L/a_{n+1}\in J. Indeed, either event forces a_{n+1}\asymp_J L, and \left|\frac{L}{a_{n+1}} -\frac{L}{a_{n+1}+x_{n+1}+y_n}\right| \le\frac{2L}{a_{n+1}^2}. Thus, whenever either of the two indicators differs from the other, both value coordinates lie in one fixed compact subset of (0,\infty), and one of them lies within O_J(L^{-1}) of an endpoint of J. Moreover, direct summation of \nu_G(a_1=q) =\frac1{\log2} \log\frac{(q+1)^2}{q(q+2)} =\frac1{\log2}\left(\frac1{q^2}+O(q^{-3})\right) shows that \begin{equation} \label{eq:rare-onepoint} \nu_G\left\{\frac{L}{a_1}\in J\right\} =\frac{|J|}{L\log2}+O_J(L^{-2}). \tag{4.11} \end{equation}

Unmarked factorial measures. We give the factorial-moment argument explicitly. Fix r, compact index-time intervals I_j, compact value intervals J_j\Subset(0,\infty), and prescribed parities. Put b_L=\lceil C_{\rm mix}\log L\rceil, with C_{\rm mix} large, and sum over distinct positions n_j\in\kappa LI_j having the prescribed parities. For the digit events A_{n,j}=\{L/a_{n+1}\in J_j\}, (4.11) gives \nu_G(A_{n,j})=\frac{|J_j|}{L\log2}+O_{J_j}(L^{-2}). If the ordered positions have successive gaps at least b_L, repeated use of (4.7) gives \begin{equation} \label{eq:long-tuple-product} \nu_G\!\left(\bigcap_{j=1}^rA_{n_j,j}\right) =\prod_{j=1}^r\nu_G(A_{n_j,j}) \left(1+O_r(\vartheta^{-b_L})\right). \tag{4.12} \end{equation} If some gap is smaller than b_L, there are only O_r(b_LL^{r-1}) possible tuples. Applying (4.7) successively with gap one bounds every such intersection by O_r(L^{-r}). Hence all short-gap tuples contribute \begin{equation} \label{eq:short-tuples} O_r(b_L/L)=o(1). \tag{4.13} \end{equation} For a value-window endpoint z>0, the replacement error in (4.10) is contained in \{\operatorname{dist}(L/a_{n+1},z)\le C/L\}, whose probability is O_z(L^{-2}), by the same direct digit summation as in (4.11). Combining this boundary event with r-1 ordinary rare events and using the gap-one bound shows that the sum of all replacement errors is O_r(L^rL^{-(r+1)})=O_r(L^{-1}).

Thus (4.12), (4.13), and the Riemann sums over the index windows prove convergence of every unmarked factorial measure in deterministic index time. The number of integers of either parity in \kappa L I_j is \frac12\kappa L|I_j|+O(1), and \kappa/(2\log2)=6/\pi^2=\lambda. Explicitly, for prescribed parities and labeled intervals as above, \begin{equation} \label{eq:unmarked-factorial-limit} \sum_{\substack{n_j\in\kappa L I_j\\ n_1,\ldots,n_r\ {\rm distinct}\\ n_j\ {\rm of\ the\ prescribed\ parity}}} \nu_G\left(\bigcap_{j=1}^r \{L\theta_{n_j}\in J_j\}\right) \longrightarrow \lambda^r\prod_{j=1}^r|I_j|\,|J_j|, \qquad \lambda=\frac6{\pi^2}. \tag{4.14} \end{equation} We now prove explicitly the uniform moment bounds that will be used below. Let \mathscr I_L be any union of a fixed number of deterministic index-time windows, so that \#\mathscr I_L=O(L), let K\Subset(0,\infty), and put R_L=\sum_{n\in\mathscr I_L}\mathbf 1_{\{L\theta_n\in K\}}. Since \theta_n=\frac1{a_{n+1}+x_{n+1}+y_n},\qquad 0<x_{n+1}+y_n<2, the event L\theta_n\in K implies c_KL\le a_{n+1}\le C_KL for suitable positive constants c_K,C_K. The latter one-digit event has Gauss probability O_K(L^{-1}). Therefore, for every fixed s\ge1, (4.9) gives \begin{align} \mathbb E_{\nu_G}(R_L)_s &=\sum_{\substack{n_1,\ldots,n_s\in\mathscr I_L\\ \text{all distinct}}} \nu_G\left(\bigcap_{j=1}^s \{L\theta_{n_j}\in K\}\right) \notag\\ &\le \sum_{\substack{n_1,\ldots,n_s\in\mathscr I_L\\ \text{all distinct}}} \nu_G\left(\bigcap_{j=1}^s \{c_KL\le a_{n_j+1}\le C_KL\}\right) \ll_{K,s}L^sL^{-s}\ll_{K,s}1. \label{eq:rare-factorial-bound}\tag{4.15}\end{align} Here (x)_s=x(x-1)\cdots(x-s+1). Since d\operatorname{Leb}/d\nu_G=\log2\,(1+x) is bounded, the same estimate holds under Lebesgue measure. Finally, \begin{equation} \label{eq:ordinary-factorial-identity} R_L^s=\sum_{j=0}^s \left\{\begin{matrix}s\\ j\end{matrix}\right\}(R_L)_j, \tag{4.16} \end{equation} where the braces are Stirling numbers of the second kind. We have therefore proved \begin{equation} \label{eq:rare-count-moments} \sup_L\mathbb E_{\nu_G}R_L^s+\sup_L\mathbb E_{\rm Leb}R_L^s<\infty \qquad(s\ge1). \tag{4.17} \end{equation}

Denominator time and the unmarked intensity. The exact identity \begin{equation} \label{eq:cf-roof} \log(Q_n+Q_{n-1}x_n) =\sum_{j=0}^{n-1}-\log x_j \tag{4.18} \end{equation} holds. The Gauss map preserves \nu_G [9, Chapter III], and its ergodicity also follows directly from (4.7). Since -\log x\in L^1(\nu_G), Birkhoff’s ergodic theorem [1] implies, locally uniformly for s\ge0, \begin{equation} \label{eq:cf-time-change} \frac{\log Q_{\lfloor\kappa Ls\rfloor}}L \longrightarrow s \quad\text{in probability}. \tag{4.19} \end{equation} Indeed, \int_0^1-\log x\,d\nu_G(x)=\frac{\pi^2}{12\log2} =\kappa^{-1}, and the two sides of (4.18) differ from \log Q_n by at most \log2. To obtain the claimed uniformity, write \mathscr S_n=\sum_{j<n}-\log x_j and \bar r=\kappa^{-1}. On every Birkhoff generic orbit, for any C<\infty and \delta>0, choose n_0 so that |\mathscr S_n-n\bar r|\le\delta n for n\ge n_0. Then \frac1L\max_{n_0\le n\le CL}|\mathscr S_n-n\bar r|\le C\delta, \qquad \frac1L\max_{n<n_0}|\mathscr S_n-n\bar r|\longrightarrow0. First let L\to\infty and then \delta\downarrow0. Together with the bounded difference above and the harmless floor in \lfloor\kappa Ls\rfloor, this proves (4.19) locally uniformly. Since Lebesgue and Gauss measure are equivalent, the same almost-sure statement holds for both.

Equations (4.11) and (4.19), together with the parity restriction in (4.5), show that each signed half has intensity \begin{equation} \label{eq:cf-intensity} \frac{\kappa}{2\log2}=\frac6{\pi^2}. \tag{4.20} \end{equation} Thus the deterministic index-time unmarked process has intensity \lambda, and (4.19) identifies the candidate denominator-time intensity. The rigorous process-level time change is carried out after marked convergence below. The uniform factorial bounds needed in that argument have already been proved above.

Transfer from Gauss to Lebesgue measure. We next make the transfer from Gauss measure to Lebesgue measure quantitative. Put w=\frac{d\operatorname{Leb}}{d\nu_G}=\log2\,(1+x),\qquad \mathscr A_b=\sigma(a_1,\ldots,a_b), and set w_L=\mathbb E_{\nu_G}(w\mid\mathscr A_{b_L}). We choose the version of this conditional expectation which is constant on each half-open depth-b_L cylinder and, at every rational endpoint, assign one of the two adjacent cylinder values. The endpoints form a null set for both measures. A depth-b continued-fraction cylinder has length \frac1{Q_b(Q_b+Q_{b-1})}\le F_{b+1}^{-2}. Since w is Lipschitz and this cylinderwise conditional average lies between the minimum and maximum of w on the cylinder, \begin{equation} \label{eq:density-cylinder-approximation} \|w-w_L\|_\infty\ll F_{b_L+1}^{-2}=o(1). \tag{4.21} \end{equation} Also w_L is uniformly bounded and \mathbb E_{\nu_G}w_L=\mathbb E_{\nu_G}w=1.

We record an aggregate version of this transfer. Suppose that X_{\mathbf n} is a collection of integrable functions, each measurable with respect to digits beginning at a position d_L, where d_L-b_L\gg L, and suppose that \sum_{\mathbf n}\mathbb E_{\nu_G}|X_{\mathbf n}|=O(1). Using w=w_L+(w-w_L), followed by (4.8), gives \begin{align} \sum_{\mathbf n} \left|\mathbb E_{\rm Leb}X_{\mathbf n} -\mathbb E_{\nu_G}X_{\mathbf n}\right| &\le \|w-w_L\|_\infty \sum_{\mathbf n}\mathbb E_{\nu_G}|X_{\mathbf n}| \notag\\ &\quad+ 2C\vartheta^{-(d_L-b_L)} \mathbb E_{\nu_G}|w_L| \sum_{\mathbf n}\mathbb E_{\nu_G}|X_{\mathbf n}| =o(1). \label{eq:aggregate-density-transfer}\tag{4.22}\end{align} For deterministic index-time intervals bounded away from zero, the earliest relevant digit position is \gg_\varepsilon L. Apply (4.22) specifically to the digit-replaced tuple functions X_{\mathbf n}=\prod_{j=1}^r \mathbf 1_{A_{n_j,j}},\qquad A_{n,j}=\{L/a_{n+1}\in J_j\}. These functions are measurable with respect to the digit future beginning at the earliest a_{n_j+1}, and their summed L^1(\nu_G)-norm is O(1) by (4.15). Thus their factorial limits are the same under Gauss and Lebesgue measure. The earlier aggregate O(L^{-1}) replacement of the exact events \{L\theta_{n_j}\in J_j\} by A_{n_j,j} also holds under Lebesgue measure, because \operatorname{Leb}(E)\le\|w\|_\infty\,\nu_G(E) for every error event E. This transfers the exact-\theta unmarked factorial limits without treating an exact-\theta event as a future digit event. Equivalently, one may first discard the initial 2b_L positions, whose expected rare count is O(b_L/L)=o(1), and retain a gap of b_L. The lower endpoint is restored by the direct cell estimate (4.52) below. This proves the same unmarked factorial limits, moment bounds, and time change for Lebesgue measure.

The following exact one-point calculation is a consistency check and is not used in the multi-point argument below. Conditional on a continued-fraction prefix through n, Lebesgue measure gives the tail x_n density \frac{1+y_n}{(1+x_ny_n)^2}\,dx_n. Under z=Lx_n/(1+x_ny_n)=L\theta_n, this becomes exactly (1+y_n)L^{-1}\,dz. Since N(Q_n\alpha-P_n)=(-1)^n\frac{N}{LQ_n}z, for every nonzero h\in\mathbb Z and compact interval J, \left|\int_J e\left(h\frac{Nz}{LQ_n}\right)\,dz\right| \le \frac{LQ_n}{2|h|N}. This tends to zero uniformly on Q_n\le N^{1-\varepsilon}. The joint independence of several such marks is the content of the oscillatory argument below.

The oscillatory marking estimate. It remains to show that the torus coordinates in (4.6) become independent Haar marks. We prove this through the Fourier coefficients of the factorial measures. Fix r, compact positive value intervals, prescribed parities, and compact deterministic index-time intervals contained in [\varepsilon,1-\varepsilon]. A general labeled r-th factorial rectangle is a sum over distinct indices, without an a priori order. Partition that sum according to the at most r! possible relative orders of the indices, and, in each part, relabel the time intervals, value intervals, parities, and Fourier frequencies accordingly. It is therefore enough to treat one such part. For ordered indices n_1<\cdots<n_r, attach the Fourier factor \begin{equation} \label{eq:joint-mark-character} \exp\left(2\pi\mathrm iN \sum_{j=1}^rh_j(Q_{n_j}\alpha-P_{n_j})\right), \qquad \mathbf h=(h_1,\ldots,h_r)\in\mathbb Z^r. \tag{4.23} \end{equation} We claim that the sum of the corresponding mixed factorial moment converges to zero whenever \mathbf h\ne0. When \mathbf h=0, the unmarked calculation above gives the required product measure.

Choose \begin{equation} \label{eq:parameter-hierarchy} 0<\Delta<\rho/100,\qquad 0<\rho<\varepsilon/100. \tag{4.24} \end{equation} Partition the deterministic time intervals into boxes of diameter at most \Delta. For each resulting ordered box configuration, write t_1<\cdots<t_r for representative times. Discard the whole configuration if |t_i-t_j|\le11\rho for some i\ne j. There are finitely many configurations for fixed \Delta,\rho; the unmarked factorial convergence already proved above shows that their aggregate mass converges to the corresponding product-Lebesgue mass. The latter is O_r(\rho), since it lies in the union of \binom r2 diagonal strips of width 22\rho+2\Delta. Thus this deletion costs O_r(\rho)+o(1). In every retained configuration, arbitrary actual times in the chosen boxes differ by more than 11\rho-2\Delta>10\rho. In particular, all ordering and separation restrictions are imposed by the boxes themselves, rather than by a later tuplewise deletion.

The unmarked factorial bounds also allow us to work on the following event. Fix once and for all C_*>3, and set \begin{equation} \label{eq:good-time-event} \mathcal G_{L,\Delta}= \left\{ \sup_{0\le n\le C_*L} \left|\frac{\log Q_n}{L}-\frac{n}{\kappa L}\right| \le\Delta\right\}. \tag{4.25} \end{equation} The probability of its complement tends to zero by (4.19). Since the rare counts have bounded moments of every fixed order, their factorial powers are uniformly integrable. Indeed, for every fixed r, (4.17) and Cauchy–Schwarz give \begin{equation} \label{eq:good-event-UI} \mathbb E\!\left[R_L^r\mathbf 1_{\mathcal G_{L,\Delta}^c}\right] \le(\mathbb ER_L^{2r})^{1/2} \mathbb P(\mathcal G_{L,\Delta}^c)^{1/2}=o(1). \tag{4.26} \end{equation} Here \mathbb E,\mathbb P may refer to either Gauss or Lebesgue measure, because (4.17) transfers as above and d\operatorname{Leb}/d\nu_G is bounded. Therefore inserting or deleting \mathcal G_{L,\Delta} changes the factorial moments by o(1). On \mathcal G_{L,\Delta}, \begin{equation} \label{eq:Q-time-box} N^{t_j-2\Delta}\le Q_{n_j}\le N^{t_j+2\Delta}. \tag{4.27} \end{equation}

Let k be the largest index for which h_k\ne0. The time separation and (4.27) give \begin{equation} \label{eq:D-dominance} D:=\sum_{j\le k}h_jQ_{n_j},\qquad |D|\asymp_{\mathbf h}Q_{n_k}. \tag{4.28} \end{equation} Indeed, for j<k, \frac{Q_{n_j}}{Q_{n_k}} \le N^{t_j-t_k+4\Delta} \le N^{-10\rho+6\Delta}=o(1), so D=h_kQ_{n_k}(1+o(1)), uniformly over the retained tuples. The integer numerator terms in (4.23) disappear, so its phase is e(ND\alpha).

For a block ending at n_\ell, let \begin{equation} \label{eq:local-denominator-event} \mathcal B_{\mathbf n,\ell} =\bigcap_{j\le\ell} \left\{N^{t_j-2\Delta}\le Q_{n_j} \le N^{t_j+2\Delta}\right\}. \tag{4.29} \end{equation} This event is measurable on the cylinder through n_\ell, and it contains \mathcal G_{L,\Delta} for every retained tuple.

Oscillatory cylinder-sum estimate. We isolate the estimate used in both the early and late cases. Fix 1\le\ell\le r, let \mathscr I_1,\ldots,\mathscr I_\ell be the integer index boxes corresponding to representative times t_1<\cdots<t_\ell, and assume that k\le\ell is the largest index for which h_k\ne0. Thus \#\mathscr I_j=O(L). For \mathbf n=(n_1,\ldots,n_\ell), put, cylinderwise, D_{\mathbf n}=\sum_{j\le k}h_jQ_{n_j},\qquad E_{\mathbf n,\ell} =\mathcal B_{\mathbf n,\ell} \cap\bigcap_{j=1}^{\ell}\{L\theta_{n_j}\in J_j\}. Then \begin{equation} \label{eq:oscillatory-cylinder-sum} \sum_{\substack{n_j\in\mathscr I_j\\ n_1<\cdots<n_\ell}} \left| \int_0^1 e\bigl(ND_{\mathbf n}(\alpha)\alpha\bigr) \mathbf 1_{E_{\mathbf n,\ell}}(\alpha)\,d\alpha \right| \ll_{\mathbf h,r,\mathbf J} L^{\ell-1}N^{\,2t_\ell-1-t_k+6\Delta}. \tag{4.30} \end{equation}

Here is the complete count. Decompose the integral for a fixed tuple into the half-open continued-fraction cylinders C of depth n_\ell. On C, all P_{n_j},Q_{n_j}, j\le\ell, and hence D_{\mathbf n}, are constant. Moreover, L\theta_{n_j} =(-1)^{n_j}LQ_{n_j}(Q_{n_j}\alpha-P_{n_j}) is affine in \alpha. The intersection of C with all value constraints is therefore either empty or a single interval I_{\mathbf n,C}, and \mathcal B_{\mathbf n,\ell} is constant on C. By the separation of the representative times and \mathcal B_{\mathbf n,\ell}, |D_{\mathbf n,C}| \ge |h_k|Q_{n_k}-\sum_{j<k}|h_j|Q_{n_j} \gg_{\mathbf h}N^{t_k-2\Delta} for all sufficiently large N, uniformly over the retained tuples. Consequently, \begin{equation} \label{eq:single-cylinder-oscillation} \left| \int_{I_{\mathbf n,C}}e(ND_{\mathbf n,C}\alpha)\,d\alpha \right| \le\frac1{\pi N|D_{\mathbf n,C}|} \ll_{\mathbf h}N^{-1-t_k+2\Delta}. \tag{4.31} \end{equation}

A finite regular continued-fraction word is determined by its reduced terminal convergent P/Q, apart from the two usual terminal expansions. Indeed, the unique canonical expansion has last digit at least two, and its only alternative replaces the terminal digit a by a-1,1; the Euclidean algorithm gives no other finite expansion. Hence the number of all finite cylinders whose terminal denominator is at most R is at most 2\sum_{Q\le R}\varphi(Q)+O(1)\ll R^2. On \mathcal B_{\mathbf n,\ell} we may take R=N^{t_\ell+2\Delta}. Once the deepest word, and hence n_\ell, is fixed, there are at most O(L^{\ell-1}) choices for n_1,\ldots,n_{\ell-1}. Notice that the cylinder count already includes the summation over n_\ell; no additional factor L is present. Multiplying this count by (4.31) gives L^{\ell-1}N^{2t_\ell+4\Delta}N^{-1-t_k+2\Delta}, which proves (4.30).

Replacing \mathbf 1_{\mathcal G_{L,\Delta}} by \mathbf 1_{\mathcal B_{\mathbf n,\ell}} changes the aggregate factorial expression only on \mathcal G_{L,\Delta}^c, hence by o(1) in view of (4.26). We use \ell=r in the early case and \ell=j_0 only for the prefix marginal in the late case; no future denominator condition is passed through a mixing estimate.

Early and late oscillatory cases. Put \sigma=(1+t_k)/2. Discard every remaining box configuration for which |t_j-\sigma|<4\rho\quad\text{for some }j>k. The unmarked product factorial measure bounds the prelimit discarded mass by O_r(\rho)+o(1). If every later event has time below \sigma-4\rho, apply (4.30) with \ell=r; its exponent is at most -8\rho+6\Delta, so the contribution is o(1).

It remains to treat the case in which a later event has time above \sigma+4\rho. Let j_0\ge k be the last event below \sigma-4\rho; include all constraints through j_0 in a prefix block. Write \mathbf n_-=(n_1,\ldots,n_{j_0}),\qquad \mathbf n_+=(n_{j_0+1},\ldots,n_r). Put s=\sigma+\rho and use the deterministic depth \begin{equation} \label{eq:deterministic-depth} m_s=\left\lceil\kappa L(s+2\Delta)\right\rceil. \tag{4.32} \end{equation} Define the genuinely prefix-measurable event \begin{equation} \label{eq:prefix-good-event} \mathcal G^-_{L,\Delta}(m_s)= \left\{ \sup_{0\le n\le m_s} \left|\frac{\log Q_n}{L}-\frac{n}{\kappa L}\right| \le\Delta \right\}. \tag{4.33} \end{equation} We have \mathcal G_{L,\Delta}\subset \mathcal G^-_{L,\Delta}(m_s)\subset \mathcal B_{\mathbf n,j_0}, and the complement of \mathcal G^-_{L,\Delta}(m_s) has probability o(1). For the second inclusion, each retained index satisfies \left|\frac{n_j}{\kappa L}-t_j\right|\le\Delta; combining this with the defining bound for \mathcal G^-_{L,\Delta}(m_s) gives |L^{-1}\log Q_{n_j}-t_j|\le2\Delta, exactly the condition in \mathcal B_{\mathbf n,j_0}. Here m_s<C_*L for all large N, by t_k\le1-\varepsilon and (4.24), so the first inclusion uses only the range present in (4.25). On \mathcal G^-_{L,\Delta}(m_s), Q_{m_s}\ge N^{s+\Delta},\qquad m_s>n_{j_0},\qquad n_{j_0+1}-m_s\ge c_\rho L. Indeed, \frac{n_{j_0}}{\kappa L} \le\sigma-4\rho+\Delta,\qquad \frac{m_s}{\kappa L}\ge\sigma+\rho+2\Delta, while \frac{n_{j_0+1}}{\kappa L} \ge\sigma+4\rho-\Delta,\qquad \frac{m_s}{\kappa L} \le\sigma+\rho+2\Delta+O(L^{-1}). The two index gaps are therefore bounded below by a positive constant times \rho L. The lower bound for Q_{m_s} follows from m_s/(\kappa L)\ge s+2\Delta and the defining error \Delta in \mathcal G^-_{L,\Delta}(m_s).

Late case: freezing the prefix. Let \mathscr N_- denote the retained prefix tuples. Define first the shallow-cylinder prefix function G^{\mathcal B}_{\mathbf n_-}(\alpha) = e\bigl(ND_{\mathbf n_-}(\alpha)\alpha\bigr) \mathbf 1_{\mathcal B_{\mathbf n,j_0}}(\alpha) \prod_{j\le j_0} \mathbf 1_{\{L\theta_{n_j}(\alpha)\in J_j\}}. The explicit cylinder estimate (4.30), now with \ell=j_0, gives \begin{align} \sum_{\mathbf n_-\in\mathscr N_-} \left|\mathbb E_{\rm Leb}G^{\mathcal B}_{\mathbf n_-}\right| &\ll L^{j_0-1}N^{2t_{j_0}-1-t_k+6\Delta} \notag\\ &\le L^{j_0-1}N^{-8\rho+6\Delta}=o(1). \label{eq:shallow-prefix-zero}\tag{4.34}\end{align}

Next replace the local denominator event by the genuinely prefix-measurable good event: G^-_{\mathbf n_-}(\alpha) = e\bigl(ND_{\mathbf n_-}(\alpha)\alpha\bigr) \mathbf 1_{\mathcal G^-_{L,\Delta}(m_s)}(\alpha) \prod_{j\le j_0} \mathbf 1_{\{L\theta_{n_j}(\alpha)\in J_j\}}. Since \mathcal G^-_{L,\Delta}(m_s)\subset \mathcal B_{\mathbf n,j_0}, the two functions agree on \mathcal G^-_{L,\Delta}(m_s). Thus \begin{align} \sum_{\mathbf n_-\in\mathscr N_-} \mathbb E_{\rm Leb} \left|G^-_{\mathbf n_-} -G^{\mathcal B}_{\mathbf n_-}\right| &\le \mathbb E_{\rm Leb}\left[ R_L^{j_0} \mathbf 1_{(\mathcal G^-_{L,\Delta}(m_s))^c}\right] \notag\\ &\le \bigl(\mathbb E_{\rm Leb}R_L^{2j_0}\bigr)^{1/2} \operatorname{Leb}\bigl((\mathcal G^-_{L,\Delta}(m_s))^c\bigr)^{1/2} =o(1). \label{eq:prefix-good-replacement}\tag{4.35}\end{align}

We now freeze this function on depth-m_s cylinders. Let \mathscr C_{m_s} be the half-open continued-fraction cylinders of depth m_s, which partition the irrational points up to a null set. We keep one fixed half-open endpoint convention and assign all rational endpoints arbitrarily; every identity in this paragraph is consequently an almost-everywhere identity, which is all that the integrals require. For every C\in\mathscr C_{m_s}, choose an irrational representative \alpha_C\in C, and write D_{\mathbf n_-,C}=\sum_{j\le k}h_jQ_{n_j}(C). The event \mathcal G^-_{L,\Delta}(m_s) is constant on every such cylinder. Define \begin{equation} \label{eq:frozen-prefix-definition} F_{\mathbf n_-}(\alpha) = \sum_{C\in\mathscr C_{m_s}} \mathbf 1_C(\alpha) \mathbf 1_{\{C\subset\mathcal G^-_{L,\Delta}(m_s)\}} e\bigl(ND_{\mathbf n_-,C}\alpha_C\bigr) \prod_{j\le j_0} \mathbf 1_{\{L\theta_{n_j}(\alpha_C)\in J_j\}}. \tag{4.36} \end{equation} This function is bounded and \sigma(a_1,\ldots,a_{m_s})-measurable.

A depth-m_s cylinder has length at most Q_{m_s}^{-2}. On a cylinder retained by \mathcal G^-_{L,\Delta}(m_s), the denominator bounds above imply \begin{equation} \label{eq:phase-freeze} \sup_{\alpha\in C} \left| e(ND_{\mathbf n_-,C}\alpha) -e(ND_{\mathbf n_-,C}\alpha_C) \right| \ll_{\mathbf h} \frac{N^{1+t_k+2\Delta}}{N^{2s+2\Delta}} =O_{\mathbf h}(N^{-2\rho}) =:\delta_N. \tag{4.37} \end{equation} Similarly, since t_j\le t_{j_0}\le\sigma-4\rho, \begin{align} \sup_{\alpha\in C} \left|L\theta_{n_j}(\alpha) -L\theta_{n_j}(\alpha_C)\right| &\le\frac{LQ_{n_j}^2}{Q_{m_s}^2} \notag\\ &\ll L N^{2(\sigma-4\rho+2\Delta) -2(\sigma+\rho+\Delta)} \le L N^{-10\rho+2\Delta} =:\eta_N \qquad(j\le j_0). \label{eq:value-freeze}\tag{4.38}\end{align}

For an interval J, write J^{(\eta)} for its closed \eta-neighborhood. The two freezing estimates give the pointwise bound \begin{align} |G^-_{\mathbf n_-}-F_{\mathbf n_-}| &\ll \delta_N \prod_{j\le j_0} \mathbf 1_{\{L\theta_{n_j}\in J_j^{(\eta_N)}\}} \notag\\ &\quad+ \sum_{i\le j_0} \mathbf 1_{\{\operatorname{dist} (L\theta_{n_i},\partial J_i)\le\eta_N\}} \prod_{\substack{j\le j_0\\j\ne i}} \mathbf 1_{\{L\theta_{n_j}\in J_j^{(\eta_N)}\}}. \label{eq:freezing-pointwise-error}\tag{4.39}\end{align} For every fixed z>0, uniformly for 0<\eta\le\min(1,z/2), \begin{equation} \label{eq:value-boundary-strip} \nu_G\{|L\theta_n-z|\le\eta\} \ll_z\frac{\eta}{L}+\frac1{L^2}. \tag{4.40} \end{equation} Indeed, on this event L\theta_n remains in a fixed compact subset of (0,\infty), and (4.10) places it inside \left\{\left|\frac{L}{a_{n+1}}-z\right| \le\eta+\frac{C_z}{L}\right\}. Direct summation of the one-digit probabilities proves (4.40).

For intersections with the other rare constraints, enlarge those constraints to their corresponding one-digit events and use (4.9). There are O(L^{j_0}) prefix tuples, so the total contribution of any one boundary coordinate is O\left( L^{j_0} \left(\frac{\eta_N}{L}+\frac1{L^2}\right) L^{-(j_0-1)} \right) =O(\eta_N+L^{-1}). The first term in (4.39) contributes O(\delta_N) by (4.15). The densities of Gauss and Lebesgue measure relative to one another are bounded, so the estimates hold under either measure. Consequently, \begin{equation} \label{eq:aggregate-freezing-error} \sum_{\mathbf n_-\in\mathscr N_-} \mathbb E_{\rm Leb}|G^-_{\mathbf n_-}-F_{\mathbf n_-}| \ll\delta_N+\eta_N+L^{-1}=o(1). \tag{4.41} \end{equation} The same calculation with all r rare factors present gives \begin{equation} \label{eq:full-aggregate-freezing-error} \sum_{\mathbf n_-,\mathbf n_+} \mathbb E_{\rm Leb}\left( |G^-_{\mathbf n_-}-F_{\mathbf n_-}| \prod_{j>j_0}\mathbf 1_{\{L\theta_{n_j}\in J_j\}}\right) =o(1). \tag{4.42} \end{equation} More precisely, after enlarging every exact rare event to its one-digit event, the phase term in this sum is O(\delta_N L^rL^{-r})=O(\delta_N), whereas a term with one prefix boundary strip is O\left( L^r\left(\frac{\eta_N}{L}+\frac1{L^2}\right) L^{-(r-1)}\right) =O(\eta_N+L^{-1}). There are only j_0\le r possible boundary coordinates. This proves (4.42) without multiplying a merely prefix-level error by the number of future tuples. Combining (4.34), (4.35), and (4.41) proves the required absolute summability, because \sum_{\mathbf n_-}|\mathbb E_{\rm Leb}F_{\mathbf n_-}| \le \sum_{\mathbf n_-}|\mathbb E_{\rm Leb}G^{\mathcal B}_{\mathbf n_-}| +\sum_{\mathbf n_-}\mathbb E_{\rm Leb} |G^-_{\mathbf n_-}-G^{\mathcal B}_{\mathbf n_-}| +\sum_{\mathbf n_-}\mathbb E_{\rm Leb} |F_{\mathbf n_-}-G^-_{\mathbf n_-}|. Thus \begin{equation} \label{eq:prefix-zero} \sum_{\mathbf n_-\in\mathscr N_-} \left|\mathbb E_{\rm Leb}F_{\mathbf n_-}\right|=o(1). \tag{4.43} \end{equation}

The frozen factor is itself dominated by enlarged one-digit rare events. In fact, if F_{\mathbf n_-}(\alpha)\ne0 and \alpha\in C, then (4.38) gives L\theta_{n_j}(\alpha)\in J_j^{(\eta_N)} \qquad(j\le j_0). Applying (4.10) on this fixed enlarged compact window, with a constant uniform in j\le j_0, yields \begin{equation} \label{eq:frozen-prefix-digit-domination} |F_{\mathbf n_-}(\alpha)| \le \prod_{j\le j_0} \mathbf 1_{\{L/a_{n_j+1}\in J_j^{(\eta_N+C_{\mathbf J}/L)}\}}. \tag{4.44} \end{equation}

Late case: prefix–future mixing. All later mark frequencies are zero. By (4.10), their exact value constraints can be replaced, with aggregate error o(1), by the one-digit constraints L/a_{n+1}\in J. To see that this remains true after multiplication by F_{\mathbf n_-}, use (4.44). If a single future indicator changes, its digit lies in an endpoint strip of probability O(L^{-2}); the frozen prefix and the remaining future coordinates are bounded by r-1 one-digit rare events of probability O(L^{-1}). Hence (4.9) gives the full aggregate bound O\bigl(L^rL^{-2}L^{-(r-1)}\bigr)=O(L^{-1}). Let H_{\mathbf n_+} denote the product of these future one-digit indicators, and let \mathscr N_+ be the retained future tuples. Then H_{\mathbf n_+} is measurable with respect to digits beginning at a_{n_{j_0+1}+1}, at distance at least c_\rho L from the prefix \sigma-field. For the fixed separated collection of index boxes now under consideration, every retained prefix tuple precedes every retained future tuple, and the boxes themselves enforce all ordering and separation conditions. Hence the retained full tuples are exactly the Cartesian product \mathscr N_-\times\mathscr N_+, which justifies the product double sums below.

We also replace the global event \mathcal G_{L,\Delta} by \mathcal G^-_{L,\Delta}(m_s). Since the former is contained in the latter, note explicitly that on any fixed digit window L/a_{n+1}\in K_0\Subset(0,\infty), 0\le \frac{L}{a_{n+1}}-L\theta_n =\frac{L(x_{n+1}+y_n)} {a_{n+1}(a_{n+1}+x_{n+1}+y_n)} \le\frac{2L}{a_{n+1}^2}=O_{K_0}(L^{-1}). Thus all digit-replaced events occurring below are contained, for large L, in a fixed enlargement of the corresponding exact \theta-windows. Take R_L here on a fixed slightly enlarged compact value window containing every exact prefix window, every enlargement in (4.44), and every digit-replaced future window. It therefore dominates every summand that occurs in this replacement, and the aggregate difference is at most \mathbb E\left[R_L^r\mathbf 1_{\mathcal G_{L,\Delta}^c}\right]=o(1) by (4.26). Together with (4.42), this shows that the late part of the original factorial Fourier coefficient is \begin{equation} \label{eq:late-frozen-sum} \sum_{\mathbf n_-\in\mathscr N_-} \sum_{\mathbf n_+\in\mathscr N_+} \mathbb E_{\rm Leb}\bigl(F_{\mathbf n_-}H_{\mathbf n_+}\bigr)+o(1). \tag{4.45} \end{equation}

Recall w=d\operatorname{Leb}/d\nu_G and w_L=\mathbb E_{\nu_G}(w\mid a_1,\ldots,a_{b_L}). Since b_L<m_s, w_LF_{\mathbf n_-} is prefix measurable. The estimates just proved, together with (4.15), give \sum_{\mathbf n_-}\mathbb E_{\nu_G}|w_LF_{\mathbf n_-}|=O(1), \qquad \sum_{\mathbf n_+}\mathbb E_{\nu_G}H_{\mathbf n_+}=O(1). The functional mixing inequality (4.8) now yields the aggregate covariance bound \begin{align} &\sum_{\mathbf n_-,\mathbf n_+} \left| \mathbb E_{\nu_G}(w_LF_{\mathbf n_-}H_{\mathbf n_+}) -\mathbb E_{\nu_G}(w_LF_{\mathbf n_-}) \mathbb E_{\nu_G}H_{\mathbf n_+} \right| \notag\\ &\qquad\le 2C\vartheta^{-c_\rho L} \left(\sum_{\mathbf n_-} \mathbb E_{\nu_G}|w_LF_{\mathbf n_-}|\right) \left(\sum_{\mathbf n_+} \mathbb E_{\nu_G}H_{\mathbf n_+}\right) =o(1). \label{eq:aggregate-late-mixing}\tag{4.46}\end{align}

Applying the same inequality to |F_{\mathbf n_-}| and H_{\mathbf n_+} shows that \sum_{\mathbf n_-,\mathbf n_+} \mathbb E_{\nu_G}|F_{\mathbf n_-}|H_{\mathbf n_+}=O(1). Thus replacing w by w_L in (4.45) costs at most \|w-w_L\|_\infty \sum_{\mathbf n_-,\mathbf n_+} \mathbb E_{\nu_G}|F_{\mathbf n_-}|H_{\mathbf n_+}=o(1). Moreover, \begin{equation} \label{eq:prefix-density-error} \sum_{\mathbf n_-} \left| \mathbb E_{\nu_G}(w_LF_{\mathbf n_-}) -\mathbb E_{\rm Leb}F_{\mathbf n_-} \right| \le \|w-w_L\|_\infty \sum_{\mathbf n_-}\mathbb E_{\nu_G}|F_{\mathbf n_-}| =o(1). \tag{4.47} \end{equation} For completeness, (4.22), applied to the future tuple indicators, also gives \begin{equation} \label{eq:future-density-error} \sum_{\mathbf n_+} \left| \mathbb E_{\rm Leb}H_{\mathbf n_+} -\mathbb E_{\nu_G}H_{\mathbf n_+} \right|=o(1). \tag{4.48} \end{equation}

Combining (4.45), (4.46), (4.47), and (4.43), the absolute value of the late contribution is at most \left(\sum_{\mathbf n_-} |\mathbb E_{\rm Leb}F_{\mathbf n_-}|\right) \left(\sum_{\mathbf n_+} \mathbb E_{\nu_G}H_{\mathbf n_+}\right)+o(1) =o(1). Finally let N\to\infty, then \Delta\downarrow0, and then \rho\downarrow0. We have proved that every nonzero joint Fourier coefficient of the marked factorial measures vanishes.

Signed extension and the point-process criterion. We have so far written the argument for positive value intervals and prescribed parity. This covers the whole signed state space. Indeed, by (4.5), (-1)^nL\theta_n>0 \quad\Longleftrightarrow\quad n\ \hbox{is even}. Thus an interval J\Subset(0,\infty) selects even indices, whereas an interval J\Subset(-\infty,0) selects odd indices after reflection to the positive interval -J. Reflection preserves Lebesgue length, so both signed halves have the same intensity. Every compact subset of \mathbb R\setminus\{0\} has positive distance from zero and is the union of its positive and negative parts. Finite disjoint decompositions therefore reduce every relatively compact signed rectangle to the two cases just proved. No rectangle accumulating at zero is needed, because such a set is not relatively compact in \mathbb R\setminus\{0\}.

Trigonometric polynomials are uniformly dense in the continuous functions on \mathbb T^r. The zero Fourier coefficient is the unmarked product factorial measure, and all other coefficients vanish. Hence, the factorial measures converge weakly in the torus variables. For a Haar-continuity rectangle, sandwich its torus indicator between continuous functions whose Haar integrals differ by an arbitrarily small amount; the uniform unmarked factorial bound controls the corresponding aggregate error. Therefore, on [\varepsilon,1-\varepsilon], all limiting factorial measures are \lambda^r\prod_{j=1}^r dt_j\,d\xi_j\,dm(u_j). We now spell out the passage from factorial measures to point processes. Let \widetilde{\mathcal Q}_N = \sum_{n\ge0} \delta_{\left(n/(\kappa L),\;(-1)^nL\theta_n,\; (-1)^nN\theta_n/Q_n\bmod1\right)} be the process in deterministic index time. On E_\varepsilon =[\varepsilon,1-\varepsilon] \times(\mathbb R\setminus\{0\})\times\mathbb T, let \mathscr R_\varepsilon be the countable semiring of relatively compact half-open rectangles whose Euclidean endpoints are rational, whose value-coordinate closures avoid zero, and whose torus endpoints belong to a fixed countable dense subset of \mathbb T. A torus arc that wraps around zero is split into two arcs. This is a dissecting, convergence-determining semiring, and every member is a continuity set for the diffuse measure \Lambda=\lambda\,dt\,d\xi\,dm.

Take pairwise disjoint B_1,\ldots,B_q\in\mathscr R_\varepsilon, and write X_{N,i}=\widetilde{\mathcal Q}_N(B_i). The factorial-measure calculation above gives, for every s_1,\ldots,s_q\ge0, \begin{equation} \label{eq:joint-count-factorial-moments} \mathbb E\prod_{i=1}^q(X_{N,i})_{s_i} \longrightarrow \prod_{i=1}^q\Lambda(B_i)^{s_i}. \tag{4.49} \end{equation} The right-hand side is the joint factorial moment of independent Poisson variables Z_i with means \Lambda(B_i).

Let T_N=\sum_iX_{N,i}. After reflecting every negative value window to its positive magnitude, T_N is bounded by a finite union of rare counts of the form used in (4.15). Consequently, for every fixed m, \sup_N\mathbb E(T_N)_m<\infty,\qquad \sup_N\mathbb ET_N^m<\infty, where the second bound follows from (4.16). In particular, the bound with one order larger makes every polynomial of a fixed degree in (X_{N,1},\ldots,X_{N,q}) uniformly integrable. Every ordinary joint moment is a finite linear combination of the joint factorial moments in (4.49); hence all ordinary joint moments converge to those of (Z_1,\ldots,Z_q). That law is moment determinate because its moment-generating function \exp\left(\sum_{i=1}^q \Lambda(B_i)(e^{t_i}-1)\right) is finite in a neighborhood of the origin. The multivariate method of moments therefore yields \begin{equation} \label{eq:joint-poisson-counts} (X_{N,1},\ldots,X_{N,q}) \Longrightarrow (Z_1,\ldots,Z_q). \tag{4.50} \end{equation}

For every finite union U of members of \mathscr R_\varepsilon, we have \Lambda(\partial U)=0, and the first factorial moment and (4.50) give \mathbb E\widetilde{\mathcal Q}_N(U)\longrightarrow\Lambda(U),\qquad \mathbb P\{\widetilde{\mathcal Q}_N(U)=0\} \longrightarrow e^{-\Lambda(U)}, respectively. Kallenberg’s mean-and-void criterion states that these two limits on the finite unions of a countable dissecting semiring of continuity sets imply convergence to the Poisson random measure [6, Theorem 4.7]. Applied to \mathscr R_\varepsilon on this locally compact second-countable space, it therefore gives vague convergence to the Poisson process with intensity \Lambda. Notice also that the mean bound supplies local tightness directly: every compact set is covered by such a finite union U, and \sup_N\mathbb E\widetilde{\mathcal Q}_N(U)<\infty, so Markov’s inequality controls the probability of more than M points uniformly as M\to\infty. Hence the marked point process converges in deterministic index time.

Time change and endpoint restoration. We give the time-change estimate at the level of test functions. Let f be continuous with compact support whose time projection lies in [\varepsilon,1-\varepsilon], and let \omega_f(\delta)= \sup_{\substack{|t-t'|\le\delta\\ \xi\in\mathbb R\setminus\{0\},\,u\in\mathbb T}} |f(t,\xi,u)-f(t',\xi,u)|. Extend f by zero outside a slightly larger compact time interval; then \omega_f(\delta)\to0 as \delta\downarrow0. Because the value projection of \operatorname{supp}f is compact in \mathbb R\setminus\{0\}, choose a compact interval K_f\Subset(0,\infty) containing \bigl\{|\xi|:(t,\xi,u)\in\operatorname{supp}f\bigr\}. Let R_L^{(f)} count all rare convergents with 0\le n\le C_*L and magnitude L\theta_n\in K_f. This contains every index-time point relevant to f. It also contains every denominator-time point relevant to f, because Q_n\ge F_{n+1} and Q_n<N imply n<C_*L for all sufficiently large N. By (4.17), \sup_L\mathbb ER_L^{(f)}+\sup_L\mathbb E(R_L^{(f)})^2<\infty.

On \mathcal G_{L,\Delta}, with \Delta<\varepsilon/2, every relevant convergent satisfies \left|\frac{n}{\kappa L}-\frac{\log Q_n}{L}\right|\le\Delta. Moreover, a point with either time coordinate in \operatorname{supp}f has the other coordinate in [\varepsilon/2,1-\varepsilon/2]. In particular Q_n<N, so the cutoff in (4.6) is present on both sides. Using the continued-fraction identification (4.6), we obtain \begin{align} \left| \langle\widetilde{\mathcal Q}_N,f\rangle -\langle\mathcal Q_N,f\rangle \right| &\le \omega_f(\Delta)R_L^{(f)} \qquad\text{on }\mathcal G_{L,\Delta}. \label{eq:test-function-time-change}\tag{4.51}\end{align} On the complement, the same difference is at most 2\|f\|_\infty R_L^{(f)}. Hence \begin{align*} \mathbb E\left| \langle\widetilde{\mathcal Q}_N,f\rangle -\langle\mathcal Q_N,f\rangle \right| &\le \omega_f(\Delta)\mathbb ER_L^{(f)} +2\|f\|_\infty \mathbb E\left[R_L^{(f)}\mathbf 1_{\mathcal G_{L,\Delta}^c}\right]\\ &\le O_f(\omega_f(\Delta))+o_{N\to\infty}(1), \end{align*} where Cauchy–Schwarz, the second-moment bound, and \mathbb P(\mathcal G_{L,\Delta}^c)=o(1) were used in the last line. First let N\to\infty, and then let \Delta\downarrow0. The converging-together argument for a countable convergence-determining family of such f’s transfers the marked process from index time to denominator time on every fixed interior layer. The endpoint layers are restored next by the direct cell bound.

It remains only to restore the two endpoint layers. Here no time-change argument is needed. Directly subdividing \alpha into the reduced rational cells about q/p gives, for every set of denominators I\subset[1,N), \begin{equation} \label{eq:endpoint-cell-count} \mathbb E\#\left\{p\in I:(p,q_p)=1, |Lp\delta_p|\le B\right\} \le \frac{2B}{L}\sum_{p\in I}\frac{\varphi(p)}{p^2} +O_B(L^{-1}). \tag{4.52} \end{equation} Indeed, for each of the \varphi(p) reduced numerators the relevant \alpha-interval has length at most 2B/(Lp^2). For p>1 these reduced cells are interior; the O_B(L^{-1}) error covers the two endpoint cells for p=1. We also have \sum_{p\le x}\frac{\varphi(p)}{p^2} =\frac6{\pi^2}\log x+O(1), as follows by inserting \frac{\varphi(p)}p=\sum_{d\mid p}\frac{\mu(d)}d and interchanging the sums. The resulting inner sum uses \sum_{m\le y}m^{-1}=\log y+O(1), while the leading coefficient is \sum_{d\ge1}\mu(d)/d^2=1/\zeta(2). Therefore the expected number of points in either discarded time layer [0,\varepsilon] or [1-\varepsilon,1], on a fixed \xi-window, is O_B(\varepsilon)+o(1). More explicitly, if f\in C_c([0,1]\times(\mathbb R\setminus\{0\})\times\mathbb T), its value projection is contained in \{|\xi|\le B_f\} for some B_f. Choose a continuous function \chi_\varepsilon:[0,1]\to[0,1] which is zero on [0,\varepsilon]\cup[1-\varepsilon,1] and one on [2\varepsilon,1-2\varepsilon]. Then f(t,\xi,u)\chi_\varepsilon(t) is a compactly supported continuous test function on an interior layer, and \mathbb E\left| \langle\mathcal Q_N,f\rangle -\langle\mathcal Q_N, f\chi_\varepsilon\rangle \right| \le \|f\|_\infty\bigl(O_{B_f}(\varepsilon)+o(1)\bigr). The limiting Poisson process satisfies the analogous bound O_f(\varepsilon) by its intensity measure. Markov’s inequality and the converging-together argument therefore restore the two layers. Letting \varepsilon\downarrow0 proves (4.2). Projection in the t-coordinate is proper because [0,1] is compact, so the point-process mapping theorem proves (4.4). ◻

Proof. For 0<r<A, restrict both point processes to r\le|\xi|\le A. The shot functional is continuous at every simple configuration with no point on the boundary, an event of limiting probability one. Lemma 4.1 and the continuous mapping theorem give convergence of the restricted shots.

It remains to let r\downarrow0. For the prelimit, a union bound and subdivision into reduced cells give \operatorname{Leb}\{\mathcal P_N((-r,r)\times\mathbb T)>0\} \le \frac{2r}{L}\sum_{p\le N}\frac{\varphi(p)}{p^2} \ll r. The limiting probability is at most 2r/\zeta(2). With probability 1-O(r), deleting |\xi|<r changes neither shot. Let r\downarrow0. ◻

Proof. The exponential formula for a Poisson process and symmetry in \xi give \begin{align*} \log\mathbb Ee^{\mathrm itX_A} &= \frac1{\zeta(2)} \int_{-A}^{A}\int_0^1 \left(e^{\mathrm itV(u)/\xi}-1\right)\,\mathrm du\,\mathrm d\xi\\ &= \frac2{\zeta(2)} \int_0^A\int_0^1 \left(\cos(tV(u)/\xi)-1\right)\,\mathrm du\,\mathrm d\xi. \end{align*} For a\ge0, \int_0^\infty\left(\cos(a/\xi)-1\right)\,\mathrm d\xi = a\int_0^\infty\frac{\cos y-1}{y^2}\,\mathrm dy =-\frac{\pi a}{2}. Moreover, 0\le1-\cos(tV(u)/\xi) \le\min\!\left(2,\frac{t^2V(u)^2}{2\xi^2}\right) \le\min(2,C_t\xi^{-2}), which is integrable for \xi>0. Dominated convergence in the nonpositive symmetric integrand therefore yields \lim_{A\to\infty}\log\mathbb Ee^{\mathrm itX_A} = -\frac{\pi|t|}{\zeta(2)}\int_0^1V(u)\,\mathrm du. Since \int_0^1V(u)\,\mathrm du=\frac1{12},\qquad \zeta(2)=\frac{\pi^2}{6}, the exponent is -|t|/(2\pi). Lévy’s continuity theorem completes the proof. ◻

Proof of Theorem 1.1. By Proposition 2.7, \frac{S_N-Y_N}{L}\longrightarrow0 \quad\text{in }L^2,\ \text{hence in probability}. For fixed A, the part of Y_N/L with |Lp\delta_p|\le A is X_{N,A}. Proposition 3.5 states precisely that the complementary part goes to zero in probability when first N\to\infty and then A\to\infty. Lemma 4.2 gives X_{N,A}\Rightarrow X_A for fixed A, and Lemma 4.3 gives X_A\Rightarrow X. Moreover, for every \eta>0, the union bound gives \begin{align*} \mathbb P\left\{\left|\frac{S_N}{L}-X_{N,A}\right|>\eta\right\} &\le \mathbb P\left\{\frac{|S_N-Y_N|}{L}>\frac\eta2\right\}\\ &\quad+ \mathbb P\left\{\left|\frac{Y_N}{L}-X_{N,A}\right|>\frac\eta2\right\}. \end{align*} Consequently, \lim_{A\to\infty}\limsup_{N\to\infty} \mathbb P\left\{\left|\frac{S_N}{L}-X_{N,A}\right|>\eta\right\}=0. The converging-together theorem [2, Theorem 3.2] now yields \frac{S_N}{\log N}\Longrightarrow X. The Cauchy law with characteristic function \exp(-|t|/(2\pi)) has continuous distribution function g(c)=\frac12+\frac1\pi\arctan(2\pi c). Weak convergence therefore gives convergence of the distribution functions at every c\in\mathbb R. The function g is nondecreasing, with limits 0 and 1 at -\infty and +\infty, respectively. ◻