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Kashin [Ka87] proves, as the introduction of Konyagin's paper [Ko94, printed p. 80] records it, that for a random polynomial $f$ of degree $n$ with independent uniform $\pm1$ coefficients the minimum modulus $m(f)=\min_{|z|=1}|f(z)|$ satisfies $m(f)\le n^{1/2}(\log n)^{-1/3}$ with probability tending to one. In Konyagin's notation $P(u)$ is the probability that $m(f)>u$; $P(\sqrt n)=0$ trivially, Littlewood [Li66] conjectured that $P(\varepsilon\sqrt n)\to0$ for every $\varepsilon>0$, and Kashin proved the conjecture through $P(n^{1/2}(\log n)^{-1/3})\to0$. Odlyzko then showed, without publishing, that $P(n^{1/3+\varepsilon})\to0$, and Konyagin's Theorem 1 gives $P(n^{-1/2+\varepsilon})\to0$ (Konyagin 1994). Kashin's bound grows with $n$, so it gives neither $m(f)<1$ nor $m(f)=o(1)$ and settles neither question of Problem 525; the first question is first answered by Konyagin's bound. The site's commentary, the introduction of Cook and Nguyen's paper (pp. 1–2; card) and the formal-conjectures statement file for the problem (its variant erdos_525.variants.kashin) instead state Littlewood's conjecture as $m(f)=o(1)$ and credit Kashin with proving it. The two accounts conflict; this page follows Konyagin's, which names the bound, and records the conflict as unresolved. Kashin's paper is not held in the library. The paper gives no earlier circulation date known to this corpus, so the page is dated by its publication year.

Covers. The bound $m(f)\le n^{1/2}(\log n)^{-1/3}$ with probability tending to one, an upper bound for the typical minimum modulus and the first progress on the second question. It settles neither the first question nor the order of $m(f)$, which Konyagin 1994, Konyagin and Schlag 1999 and Cook and Nguyen 2021 determine.

Depends on. Nothing in this wiki; the result rests on the published paper cited above.

Acceptance. Refereed: the paper appeared in Vestnik Moskovskogo Universiteta, Seriya I, Matematika, Mekhanika (1987), 40–46, 105, and Konyagin's refereed paper of 1994 records the theorem in its introduction. The site's curator credits Kashin with Littlewood's conjecture in the form $m(f)=o(1)$, which Konyagin's account contradicts, so that credit is not listed as reviewed. No formal proof of this bound is held or audited in this repository, so no formalized evidence is listed.