Problem 524

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claims/: The 2 claim pages of Problem 524, one per claimant's result; the problem's standing derives from them.

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Statement. For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all $t\in(0,1)$) for

$$M_n(t)=\max_{x\in [-1,1]}\left\lvert \sum_{k\leq n}(-1)^{\epsilon_k(t)}x^k\right\rvert?$$

Formulation. The site's statement was corrected on 27 December 2025, after a thread comment, to the signs $(-1)^{\epsilon_k(t)}$ and the interval $[-1,1]$; the earlier text used the digits $\epsilon_k(t)$ themselves and the interval $[0,1]$, for which the maximum is just the digit sum. Erdős [Er61, p. 253] defines $M_n(t)$ as the maximum over $\lvert x\rvert\le1$, says that the law of the iterated logarithm gives its upper bound, asks for its exact lower bound, which "seems very difficult", and attributes the problem to Salem and Zygmund [SaZy54]. The question is read as Erdős reads it and as Letwin and Sawhney read it: for almost every $t$, the signs are independent Rademacher variables, and the question asks for the almost sure lower envelope of $M_n(t)$, the upper envelope being Salem and Zygmund's.

Status. OPEN (the site's label; page last edited 27 December 2025). The derived standing, claimed and answered, departs from the label because of a pending full claim: Letwin–Sawhney 2026, an arXiv preprint announced on the thread on 2026-04-24, determines the almost sure lower envelope, and with Salem and Zygmund's upper envelope that answers the question. A partial claim precedes it: Chojecki 2026, a note posted on the thread on 2026-01-30, reproves the upper envelope and finds the lower-envelope scale along a sparse subsequence. Neither is refereed, registered on the site's proof-claims tab, or accepted by the site.

Source. erdosproblems.com/524, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #524, https://www.erdosproblems.com/524.

References.

• [Ch26] Chojecki, P., Maximum of random $\pm1$ polynomials on $[-1,1]$: a.s. order and the lower envelope. Note dated 30 January 2026, https://www.ulam.ai/research/erdos524.pdf.
• [Er61] Erdős, P., Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221-254; Part V, item 3, p. 253. Library home: erdos_1961_unsolved_problems.
• [LeSa26] Letwin, B. and Sawhney, M., On the maxima of Littlewood polynomials on $[-1,1]$. arXiv:2604.19294 (2026). Library home: letwin_2026_maxima_littlewood_polynomials_1_1.
• [SaZy54] Salem, R. and Zygmund, A., Some properties of trigonometric series whose terms have random signs. Acta Math. (1954), 245-301.

Formalization. None recorded: no formal-conjectures statement file exists, and the community database lists the problem as unformalized.

Current assessment

The site labels the problem OPEN (page last edited 27 December 2025) and its remarks record two bounds. Salem and Zygmund [SaZy54, Theorem (6.1.1)] proved the upper envelope: almost surely $\limsup_n M_n(t)/\sqrt{2n\log\log n}=1$. The remarks credit Chung with $M_n(t)\ll(n/\log\log n)^{1/2}$ for infinitely many $n$, almost surely, and Erdős with the unpublished result that $M_n(t)/n^{1/2-\epsilon}\to\infty$ for every $\epsilon>0$. Neither gets a claim page: Erdős [Er61, p. 253] reports the first as implied by a theorem of Chung and states the second as his own unpublished result, and no printed proof of either is cited anywhere.

Two claims postdate the page. Chojecki's note [Ch26], posted on the thread on 2026-01-30 and written, as the post says, with GPT 5.2 and a small input from Gemini and Grok, reproves the Salem–Zygmund upper envelope and shows that along $n_m=\lfloor e^{m^3}\rfloor$ the minimal values of $M_n(t)$ occur at the scale $\sqrt n\exp(-\Theta((\log\log n)^{1/3}))$, with the constant pinned between two values taken from the Gao–Li–Wellner small-ball bounds; it does not determine the lower envelope, as its conclusion says. The same day the site's curator wrote that he would view the problem as unresolved until an expert confirmed the argument or a formal proof existed, and Mehtaab Sawhney replied that the proof is correct and is what he had sketched on the thread, adding that neither the note nor his coming result with Letwin pins the answer down as precisely as Salem and Zygmund might have wanted. That result is the pending full claim [LeSa26]: almost surely

$$\liminf_{n\to\infty}\frac{\log\bigl(M_n(t)/\sqrt n\bigr)}{(\log\log n)^{1/3}} =-\Bigl(\frac{3\pi^2}{4}\Bigr)^{1/3},$$

the lower envelope through the small-ball probability of the Gaussian process $\int_0^1e^{-st}\,dB_s$ and the leading constant of that probability. With the upper envelope this determines both envelopes of $M_n(t)$, the answer Erdős asked for. The preprint is not refereed, no proof claim is registered on the site's proof-claims tab, and neither argument is compiled or reviewed in this wiki; the site's page predates both postings.

Search scope: the site's problem page as exported (last edited 27 December 2025), its discussion thread (11 comments) and proof-claims tab (none), the community database entry (open, unformalized), the arXiv record of the Letwin–Sawhney preprint and the note's own page, all read 2026-10-07; no formal-conjectures statement exists and no OpenAI release item names this problem. MathSciNet and zbMATH were not searched and X was not used.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• letwin_2026_maxima_littlewood_polynomials_1_1
• letwin_2026_maxima_littlewood_polynomials_1_1 / theorem_1_1
• letwin_2026_maxima_littlewood_polynomials_1_1 / theorem_1_2
• letwin_2026_maxima_littlewood_polynomials_1_1 / theorem_2_7