Measure growth in compact semisimple Lie groups and the Kemperman Inverse Problem

TL;DR

This paper establishes a quantitative lower bound on the measure of the square of a set in compact semisimple Lie groups and solves the Kemperman Inverse Problem from 1964, advancing understanding of measure growth and inverse problems in group theory.

Contribution

It provides a universal lower bound for measure growth in compact semisimple Lie groups and resolves the longstanding Kemperman Inverse Problem.

Findings

Established a measure growth inequality with an explicit constant.

Resolved the Kemperman Inverse Problem from 1964.

Extended results to connected compact groups without toric quotients.

Abstract

Suppose $G$ is a compact semisimple Lie group, $\mu$ is the normalized Haar measure on $G$, and $A, A^2 \subseteq G$ are measurable. We show that $$\mu(A^2)\geq \min\{1, 2\mu(A)+\eta\mu(A)(1-2\mu(A))\}$$ with the absolute constant $\eta>0$ (independent from the choice of $G$) quantitatively determined. We also show a more general result for connected compact groups without a toric quotient and resolve the Kemperman Inverse Problem from 1964.