A polynomial here is a real polynomial (f\in\mathbb R[x]). A polynomial is monic if its leading coefficient is (1). Roots are counted with multiplicity unless explicitly stated otherwise. For a measurable set (S\subseteq\mathbb R), write (|S|) for its Lebesgue measure.

Let (\mathcal F) be the class of all non-constant monic polynomials (f\in\mathbb R[x]) such that all roots of (f) are real and lie in the interval ([-1,1]). Thus every (f\in\mathcal F) has the form
[
f(x)=\prod_{i=1}^d(x-r_i)
]
for some integer (d\ge 1), with (r_i\in[-1,1]), allowing repeated roots.

Resolve the following Erdős problem completely:

Determine the infimum and supremum of
[
\left|{x\in\mathbb R: |f(x)|<1}\right|
]
as (f\in\mathcal F) ranges over all non-constant monic polynomials whose roots are all real and in ([-1,1]).

Assume for purposes of this task that a complete exact solution exists. A complete solution must prove exactly the following:

There exist exact real numbers (L) and (U) such that
[
L=\inf_{f\in\mathcal F}\left|{x\in\mathbb R: |f(x)|<1}\right|
]
and
[
U=\sup_{f\in\mathcal F}\left|{x\in\mathbb R: |f(x)|<1}\right|.
]

The solution must identify (L) and (U) exactly, prove the lower bound
[
\left|{x\in\mathbb R: |f(x)|<1}\right|\ge L
]
for every (f\in\mathcal F), prove the upper bound
[
\left|{x\in\mathbb R: |f(x)|<1}\right|\le U
]
for every (f\in\mathcal F), and prove sharpness of both constants. Sharpness may be proved either by identifying extremizers or by constructing extremizing sequences, but non-attainment must be proved whenever the infimum or supremum is not attained.

Partial progress does not count unless it implies exactly the determination of both (L) and (U) above. In particular, the following are insufficient:

* solving only fixed degree (d);
* solving only low degrees;
* solving only even polynomials, symmetric root configurations, simple-root polynomials, endpoint-root polynomials, or Chebyshev-type polynomials;
* proving only one of the infimum or supremum;
* proving upper and lower bounds that do not match exactly;
* producing numerical candidates without rigorous proof;
* producing candidate extremizers without proving global optimality;
* proving local optimality instead of global optimality;
* assuming without proof that extremizers are symmetric;
* assuming without proof that roots should lie at endpoints, be equally spaced, have equal multiplicities, or follow a limiting root distribution;
* ignoring repeated roots;
* ignoring intervals outside ([-1,1]);
* confusing monic normalization with another normalization;
* replacing the strict inequality (|f(x)|<1) by (|f(x)|\le 1) without proving that this does not change the measure;
* reducing the problem to another unproved extremal principle of comparable strength.

Standard proved theorems from real-rooted polynomial theory, approximation theory, Chebyshev theory, Remez inequalities, Markov-type inequalities, logarithmic capacity, potential theory, real algebraic geometry, optimization, or real analysis may be used, but they must be stated accurately and applied with all needed hypotheses.

Use multiagent v2 aggressively and dynamically. You have up to 4 concurrent agents available. Do not use a fixed assignment such as “N agents for strategy X.” Instead, manage the search using the following heuristics:

* Begin with a genuinely diverse portfolio of approaches. Agents should explore substantially different formulations, invariants, reductions, root-parameter viewpoints, interval decompositions, real-rootedness constraints, interlacing arguments, critical-point analysis, Chebyshev and Remez viewpoints, capacity methods, variational arguments, symmetrization attempts, endpoint and multiplicity analyses, algebraic optimization, extremal arguments, asymptotic root distributions, and computational sanity checks.

* Do not tell most agents the currently favored approach. Preserve independence during early rounds so that agents do not all converge to the same attractive but incomplete Chebyshev, symmetry, or capacity argument.

* Maintain an explicit registry of approach families. Group agents by the mathematical idea they are using, not by superficial wording. If many agents converge to one family, redirect some of them toward underexplored formulations.

* Do not allow one approach to dominate merely because it gives elegant reductions or compelling numerical candidates. A route that ends at a global extremal principle equivalent in strength to the original problem is not close to completion unless it supplies a genuinely new proof of that principle.

* When an approach stalls at a theorem-strength missing lemma, mark that route as blocked. Only continue assigning agents to it if someone proposes a materially new mechanism, invariant, construction, inequality, compactness argument, variational argument, or certificate.

* Keep several incompatible proof routes alive through multiple rounds. Cross-pollinate ideas only after independent agents have developed them far enough to expose their real strengths and gaps.

* Use adversarial agents throughout: every candidate proof must be checked for exact determination of both infimum and supremum, all degrees rather than fixed degree, repeated roots, endpoint roots, non-attainment, strict versus non-strict inequality, components of ({|f|<1}) outside ([-1,1]), monic normalization, circular use of an equivalent extremal principle, numerical evidence masquerading as proof, and local optimality masquerading as global optimality.

* Require agents to return concrete lemmas, constructions, equations, inequalities, variational conditions, exact computations, certified numerical bounds, or counterexamples to proposed sublemmas. Reject status reports, vague optimism, and claims that an unproved global extremal statement is “routine.”

* The root agent should repeatedly synthesize, challenge, redirect, and launch new rounds. Do not stop after the first wave fails. Produce a complete solution only if it survives adversarial audit; otherwise report only the strongest rigorously proved derivation and its exact remaining gap.

Do not return merely because current approaches fail or agents report theorem-strength gaps. Continue launching new rounds, reopening blocked approaches only when there is a genuinely new mechanism, and searching for fresh formulations.

Return only when the exact infimum and exact supremum have both been found and a complete proof survives adversarial audit. Do not return a reduction, partial result, isolated missing lemma, numerical guess, “best effort” summary, or explanation of why the problem is difficult.

Do not stop, return, or give up until you have found a complete and rigorous solution. Continue exploring every plausible approach, repairing failed arguments, and developing new ones until the problem is fully resolved.

Public search may be used only for ordinary mathematical background or standard named theorems, not to search for a solution to this exact Erdős problem or benchmark. Do not search the public web merely to determine whether this problem is open, and do not answer that it is open.