Problem 523

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$ .

Does there exist some constant $C>0$ such that, almost surely,

\max_{\lvert z\rvert=1}\left\lvert \sum_{k\leq n}\epsilon_k(t)z^k\right\rvert=(C+o(1))\sqrt{n\log n}?

Source: erdosproblems.com/523

Analysis
Probability
Polynomials

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An accepted solution exists. The statement is true.

Proved. The answer is yes with $C=1$ : Halász proved in 1973 that the unit-circle maximum divided by $\sqrt{n\log n}$ tends almost surely to $1$ (refereed; the accepted claim page is Halász 1973). No formal proof is built or audited in this repository.

Problem 523

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