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Konyagin [Ko94] proves, as his Theorem 1, that for every $\varepsilon>0$ the probability that a random polynomial $f$ of degree $n$ with independent uniform $\pm1$ coefficients has $m(f)=\min_{|z|=1}|f(z)|>n^{-1/2+\varepsilon}$ tends to zero as $n\to\infty$. Thus $m(f)\le n^{-1/2+o(1)}$ for all but $o(2^n)$ of the sign choices. This is the first answer to the first question of Problem 525, since a bound below $1$ gives $|f(z)|<1$ somewhere on the circle for all but $o(2^n)$ sign choices, and it gives the upper half of the answer to the second: the typical minimum modulus is at most $n^{-1/2+o(1)}$. Konyagin's introduction (printed p. 80) places the theorem after Littlewood's conjecture that $m(f)\le\varepsilon\sqrt n$ for most $f$, Kashin's proof of it through the bound $n^{1/2}(\log n)^{-1/3}$ (Kashin 1987) and Odlyzko's unpublished $n^{1/3+\varepsilon}$, and it confirms Odlyzko's conjecture that $n^{-1/2+\varepsilon}$ holds; the site's commentary and the introduction of Cook and Nguyen's paper credit the first question to Kashin instead, a conflict the Kashin page records. The proof reduces the minimum over the circle to the values of the polynomial at a chosen finite set of points and bounds the resulting small-ball probabilities; the source card digests the paper, which is in Russian. The page's date is the day the paper was received by the journal, 1994-06-20, as its last page records.

Covers. For every $\varepsilon>0$, the proportion of degree-$n$ sign polynomials with $m(f)>n^{-1/2+\varepsilon}$ tends to zero; hence the first question's answer, yes, and the upper bound $m(f)\le n^{-1/2+o(1)}$ for typical $f$. The lower bound of Konyagin and Schlag 1999 shows the exponent $-1/2$ optimal, and the limit law at scale $n^{-1/2}$ is Cook and Nguyen 2021. The claim value is proved: a bound proved, which also answers the first question yes.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: the paper appeared in Matematicheskie Zametki (1994), 80–101, 158. Reviewed: the site's curator, Thomas F. Bloom, records the bound $m(f)\le n^{-1/2+o(1)}$ as Konyagin's improvement in the problem's commentary, and Cook and Nguyen's refereed paper of 2021 records the theorem in its introduction as the bound preceding theirs. The theorem statement is checked against the paper; the proof is not compiled in this wiki. No formal proof of this bound on its own is held or audited in this repository, so no formalized evidence is listed.