The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and F\"uredi

TL;DR

This paper establishes a new probability bound for random sums in high dimensions and proves a longstanding conjecture by Frankl and F"uredi, extending classical Littlewood-Offord results to higher dimensions.

Contribution

It introduces a novel bound on the probability of high-dimensional random sums and confirms a conjecture that generalizes classical Littlewood-Offord theorems.

Findings

New probability bound for high-dimensional sums

Proof of Frankl and F"uredi's conjecture from 1988

Extension of Littlewood-Offord theorem to high dimensions

Abstract

We give a new bound on the probability that the random sum $\xi_1 v_1 +...+ \xi_n v_n$ belongs to a ball of fixed radius, where the $\xi_i$ are iid Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an application, we prove a conjecture of Frankl and F\"uredi (raised in 1988), which can be seen as the high dimensional version of the classical Littlewood-Offord-Erd\H os theorem.