Problem 524

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?

Source: erdosproblems.com/524

Analysis
Probability
Polynomials

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A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.

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.

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: Erdős 1961.
LeSa26: Letwin, B. and Sawhney, M., On the maxima of Littlewood polynomials on $[-1,1]$ . arXiv:2604.19294 (2026). Library home: Letwin 2026.
SaZy54: Salem, R. and Zygmund, A., Some properties of trigonometric series whose terms have random signs. Acta Math. (1954), 245-301.

Problem 524

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