Non-abelian Littlewood-Offord inequalities

TL;DR

This paper introduces the first non-abelian analogue of the Littlewood-Offord inequality, providing a sharp anti-concentration result for products of independent random variables, expanding the scope of classical sum-based inequalities.

Contribution

It presents a novel non-abelian anti-concentration inequality for products of independent random variables, extending classical results to a new algebraic setting.

Findings

Establishes a sharp anti-concentration inequality for non-abelian products

Generalizes classical Littlewood-Offord results to non-commutative groups

Provides tools for analyzing randomness in non-abelian algebraic structures

Abstract

In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.