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Claim. The note, 13 pages dated 30 January 2026 and posted on the site's thread the same day, says in its post that it was made with GPT 5.2, with a small input from Gemini and Grok, following Mehtaab Sawhney's thread remarks. It writes $M_n=\max_{x\in[-1,1]}\lvert\sum_{k\le n}a_kx^k\rvert$ with independent Rademacher signs $a_k$, which for almost every $t$ are the signs $(-1)^{\epsilon_k(t)}$ of Problem 524. Theorem 6: almost surely $\limsup_nM_n/\sqrt{2n\log\log n}=1$, by Abel summation, which bounds $M_n$ by the running maxima of the two random walks $\sum_{j\le k}a_j$ and $\sum_{j\le k}(-1)^ja_j$, and the law of the iterated logarithm in Kolmogorov's and Chung's forms. This is Salem and Zygmund's Theorem (6.1.1) [SaZy54] reproved. Theorem 18: with $n_m=\lfloor e^{m^3}\rfloor$, $\alpha_-=(6\bar c)^{-1/3}$ and $\alpha_+=(6\underline c)^{-1/3}$, where $\underline c\le\bar c$ are the constants of the two-sided Gao–Li–Wellner small-ball bounds $e^{-\bar c\lvert\log\epsilon\rvert^3}\le\mathbb P(\sup_{u\ge0}\lvert Y(u)\rvert\le\epsilon)\le e^{-\underline c\lvert\log\epsilon\rvert^3}$ for $Y(u)=\int_0^1e^{-us}\,dB(s)$, almost surely

$$\alpha_-\le\limsup_{m\to\infty} \frac{\log\bigl(\sqrt{n_m}/M_{n_m}\bigr)}{(\log\log n_m)^{1/3}}\le\alpha_+ .$$

The route writes $x=\pm e^{-u/n}$, so that $M_n/\sqrt n$ is the supremum of two Laplace-type processes of the walks, couples the pair to two independent copies of $Y$ by a two-dimensional strong invariance principle, transfers the small-ball estimate to $M_n/\sqrt n$ at scales $\epsilon_n\gg n^{-1/2}$, and applies both Borel–Cantelli lemmas to events that depend on disjoint blocks of coefficients. Section 5.1 also states the bound $\liminf_n(\log\log n/n)^{1/2}M_n\le\pi/\sqrt8$ from Chung's law for the running maximum; its derivation takes the lower limit of a maximum to be at most the maximum of the lower limits, which is false in general, so this page does not record that bound as proved. The abstract says the note answers Problem 524; its conclusion says that identifying the exact constant in an almost sure limit theorem for $\log(\sqrt n/M_n)/(\log\log n)^{1/3}$ would need an exact small-ball constant for $Y$ and a full-sequence dependence analysis, which it calls the main outstanding step.

Covers. For the whole sequence, almost surely $M_n(t)\le\sqrt n\exp(-\alpha(\log\log n)^{1/3})$ for infinitely many $n$, for every $\alpha<\alpha_-$, since the subsequence events occur infinitely often; the two-sided scale $\sqrt n\exp(-\Theta((\log\log n)^{1/3}))$ holds only along $n_m$. The note does not determine the lower envelope of $M_n(t)$, which is the full claim on Letwin–Sawhney 2026.

Depends on. No page of this wiki.

Standing. Claimed. On 2026-01-30 the site's curator, Thomas Bloom, wrote on the thread that proofs generated by language models are prone to subtle mistakes, especially when they invoke tools from the literature, and that Bloom would view the problem as unresolved until an expert confirmed the argument or a formal proof existed; the author replied that they were running the note through Aristotle for a possible formalization, of which none is recorded. Mehtaab Sawhney replied the same day that the proof is correct and is what Sawhney had suggested in a sketch on the thread, the sketch being implemented with the Komlós–Major–Tusnády coupling, and later that neither this note nor Sawhney's coming result with Brayden Letwin pins the answer down as precisely as Salem and Zygmund might have wanted. These posts are not read as acceptance: the site labels the problem OPEN (page last edited 27 December 2025, before the postings), no proof claim is registered on the site's proof-claims tab, and Sawhney checked an argument Sawhney had suggested while co-authoring a competing result. Nothing is compiled or reviewed in this wiki.