Erdős Problem #524 — For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$).

The problem asks for the almost-sure size of the maximum of a random polynomial on the interval from minus one to one. The lower-envelope question was raised by Salem and Zygmund and later reiterated by Erdős.

Known results

• Salem and Zygmund: almost surely, the upper envelope satisfies

$$\limsup_{n\to\infty}\frac{M_n}{\sqrt{n\log\log n}}=\sqrt{2}.$$

Letwin–Sawhney result (2026)

Brayden Letwin and Mehtaab Sawhney determine the lower envelope through Gaussian-process small-ball estimates and obtain

$$\liminf_{n\to\infty}\frac{\log(M_n/\sqrt n)}{(\log\log n)^{1/3}}=-\frac{3\pi}{4^{1/3}}.$$

Thus the minimum values occurring infinitely often have scale $\sqrt n\exp\!\bigl(-\Theta((\log\log n)^{1/3})\bigr)$. The accompanying discussion says the argument was completed with assistance from GPT 5.2, with a small input from Gemini/Grok.

Current status (as of June 2026): The almost-sure upper and lower envelopes are settled by the Letwin–Sawhney paper.