A non-uniform Littlewood-Offord inequality

TL;DR

This paper extends the classical Littlewood-Offord inequality to a non-uniform setting, providing optimal bounds for the probability that a weighted sum of Rademacher variables equals a specific vector, based on the vector's length.

Contribution

It introduces a non-uniform version of the Littlewood-Offord inequality and derives the optimal probability bounds depending on the target vector's length.

Findings

Derived the optimal bounds for probability of sum equaling a vector based on its length.

Extended classical Littlewood-Offord results to a non-uniform setting.

Provides theoretical tools for analyzing sums of random vectors with non-uniform targets.

Abstract

Consider a sum $S_n=v_i\varepsilon_1+\cdots+v_n\varepsilon_{n}$, where $(v_i)^{n}_{i=1}$ are non-zero vectors in $\mathbb{R}^{d}$ and $(\varepsilon_i)^{n}_{i=1}$ are independent Rademacher random variables (i.e., $~{\mathbb{P}(\varepsilon_{i}=\pm 1)=1/2}$). The classical Littlewood-Offord problem asks for the best possible upper bound for $~{\sup_{x}\mathbb{P}(S_n = x)}$. In this paper we consider a non-uniform version of this problem. Namely, we obtain the optimal bound for $\mathbb{P}(S_n = x)$ in terms of the length of the vector $x\in \mathbb{R}^d$.