Geometric and o-minimal Littlewood-Offord problems

TL;DR

This paper extends the Littlewood-Offord problem to geometric and o-minimal structures, showing bounds on probabilities for sums of random signs falling into definable sets, with inverse results.

Contribution

It introduces bounds for the probability that a random signed sum lies in o-minimal definable sets, generalizing classical results to geometric and logical structures.

Findings

Probability bound of n^{-1/2+o(1)} for sums in definable sets

Applicable to algebraic hypersurfaces and similar structures

Provides an inverse theorem in the o-minimal setting

Abstract

The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,\dots,a_n\in \mathbb{R}^d$, any $x\in \mathbb{R}^d$, and uniformly random $(\xi_1,\dots,\xi_n)\in\{-1,1\}^n$, we have $\Pr(a_1\xi_1+\dots+a_n\xi_n=x)=O(n^{-1/2})$. In this paper we show that $\Pr(a_1\xi_1+\dots+a_n\xi_n\in S)\le n^{-1/2+o(1)}$ whenever $S$ is definable with respect to an o-minimal structure (for example, this holds when $S$ is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.