Problem 395

If $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i\rvert=1$ then is it true that the probability that

\lvert \epsilon_1z_1+\cdots+\epsilon_nz_n\rvert \leq \sqrt{2},

where $\epsilon_i\in \{-1,1\}$ uniformly at random, is $\gg 1/n$ ?

Source: erdosproblems.com/395

Analysis

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. The site shows PROVED (LEAN), a catalog label explained under Formalization. Two accepted full claims answer the question: Beck's 1983 theorem, a refereed journal paper recorded as [HJNS24] and [HPS25] quote it, whose planar case is the question; and Theorem 1.1 of He, Juškevičius, Narayanan and Spiro [HJNS24], an arXiv preprint credited by the site's curator.

CaCa11: Carnielli, Walter and Carolino, Pietro K., Adjusting a conjecture of Erdős. Contrib. Discrete Math. 6 (2011), no. 1, 154-159.
Er45: Erdős, P., On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51 (1945), 898-902; p. 902, conjecture (1). Library home: Erdős 1945.
HJNS24: X. He, T. Juškevičius, B. Narayanan, and S. Spiro, The Reverse Littlewood-Offord problem of Erdős. arXiv:2408.11034 (2024). Library home: He et al. 2024 (arXiv v3, 30 December 2024, whose PDF title is On the reverse Littlewood–Offord problem of Erdős).
HPS25: L. Hollom, J. Portier, and V. Souza, Double-jump phase transition for the reverse Littlewood–Offord problem. arXiv:2503.24202v1 (31 March 2025). Library home: Hollom, Portier and Souza 2025.
HS25: L. Hollom and G. B. Sorkin, Reverse Littlewood–Offord problems with parity conditions. arXiv:2510.05044v1 (6 October 2025). Library home: Hollom and Sorkin 2025.

Problem 395

Status

Proof claims

Discussion