Measure of $\{|f|<1\}$ for real-rooted monic polynomials in $[-1,1]$: pin down the infimum (Erdős #1038)
Statement
Determine the infimum and supremum of the Lebesgue measure $$\lvert\{x\in\mathbb{R} : \lvert f(x)\rvert < 1\}\rvert$$ as $f\in\mathbb{R}[x]$ ranges over all non-constant monic polynomials all of whose roots are real and lie in the interval $[-1,1]$.
Acceptance. FULLY RESOLVES: exact determination of the infimum (the supremum is reportedly settled at $2\sqrt{2}$ — confirm and cite): a closed-form or explicitly characterized value $I$, a proof that every admissible polynomial has measure $\geq I$, and a construction (or limiting family) achieving measure $\to I$; machine-checkable proof preferred, else a full written proof. ADVANCES: (a) an explicit admissible polynomial (degree, roots and multiplicities stated exactly) whose sub-level-set measure is certified — by exact rational/algebraic computation or interval arithmetic — to be strictly below the best upper bound on the infimum stated in the background; (b) a proven lower bound strictly above the best lower bound stated in the background; or (c) a self-contained proof of the supremum value $2\sqrt{2}$ suitable for the curated record. Any claimed record must beat the CURRENT best on the site's discussion at submission time, not merely the values quoted here. Deliver the construction with certification code and output, or the proof file.
Background
A problem of Erdős, Herzog, and Piranian [EHP58, p.131]; listed as open on erdosproblems.com/1038 (fetched 2026-07-13, status 'open', tagged 'analysis'). EHP proved the measure is at most $2\sqrt{2}$ when all roots lie in $\{-1,1\}$ and conjectured this is the best possible upper bound in general. On the infimum side they observed the value is below $2$, witnessed by $f(x)=(x+1)(x-1)^m$ for $m\geq 3$; if the roots are instead allowed in $[-2,2]$ the infimum drops to $0$ (small perturbations of Chebyshev polynomials), though not too fast: EHP conjectured a lower bound $n^{-c}$ in that regime, which Pommerenke [Po61] proved, showing the set even contains an interval of width $\gg n^{-4}$. This problem is currently very active: the site's summary of a long comment thread (141 comments; participants include Terence Tao, and several users report it tractable) gives the current records as $$2^{4/3}-1\approx 1.519 \leq \inf \leq 1.835\cdots$$ with the supremum now settled at $\sup=2\sqrt{2}\approx 2.828$ per that discussion — so the live target is the infimum, whose window $[1.519,\,1.835]$ has been narrowing through explicit constructions and arguments in the thread. NOTE: the curated remarks already reproduce these bounds with a pointer to the (very active) comment thread, and the records may move; re-verify the current best on the site before claiming an improvement. The attacker's tool: for the upper bound on the infimum, explicit constructions — optimize root locations/multiplicities in $[-1,1]$ numerically, compute the sub-level-set measure exactly by root isolation, and certify with interval arithmetic; the lower-bound side needs proof, but of a concrete extremal-polynomial kind well suited to certified computation.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1038 (T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.