Bound for the maximal probability in the Littlewood-Offord problem

TL;DR

This paper establishes bounds on the maximum probability in the Littlewood-Offord problem by linking it to the concentration functions of symmetric infinitely divisible distributions, providing new estimation techniques.

Contribution

It introduces a novel connection between the Littlewood-Offord problem and symmetric infinitely divisible distributions, enabling improved probability bounds.

Findings

Values of concentration functions at zero can be estimated using symmetric infinitely divisible distributions.

The approach involves Lév́y spectral measures related to weights in sums.

Provides a new method for analyzing concentration probabilities in probabilistic combinatorics.

Abstract

The paper deals with studying a connection of the Littlewood--Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the values at zero of the concentration functions of weighted sums of i.i.d. random variables may be estimated by the values at zero of the concentration functions of symmetric infinitely divisible distributions with the L\'evy spectral measures which are multiples of the sum of delta-measures at $\pm$weights involved in constructing the weighted sums.