The Reverse Littlewood--Offord problem of Erd\H{o}s

TL;DR

This paper proves a sharp probabilistic bound for sums of random vectors in two dimensions, resolving Erdős's longstanding conjecture and extending results to higher dimensions with polynomial bounds.

Contribution

It establishes the first proof of Erdős's conjecture on the Littlewood–Offord problem for two dimensions, with sharp bounds and extensions to higher dimensions.

Findings

Probability bound of c/n for vector sums in 2D

Sharpness of the bound with constant √2

Polynomial bounds in higher dimensions

Abstract

Let $\epsilon_{1},\ldots,\epsilon_{n}$ be a sequence of independent Rademacher random variables. We prove that there is a constant $c>0$ such that for any unit vectors $v_1,\ldots,v_n\in \mathbb{R}^2$, $$\Pr\left[||\epsilon_1 v_1+\ldots+\epsilon_n v_n||_2 \leq \sqrt{2}\right]\geq \frac{c}{n}.$$ This resolves the only remaining conjecture from the seminal paper of Erd\H{o}s on the Littlewood--Offord problem, and it is sharp both in the sense that the constant $\sqrt{2}$ cannot be reduced and that the magnitude $n^{-1}$ is best possible. We also prove polynomial bounds for the analogous problem in higher dimensions.