Minimal and nearly minimal measure expansions in connected unimodular groups

TL;DR

This paper characterizes when measure expansions in connected unimodular groups are minimal or nearly minimal, solving a longstanding question and confirming conjectures, with applications to Lie groups and measure expansion gaps.

Contribution

It provides a complete characterization of equality cases and near-equality bounds for measure expansions in connected unimodular groups, confirming conjectures and extending previous results.

Findings

Complete characterization of equality in measure expansion inequality.

Quantitative bounds for nearly minimal measure expansions.

Application to measure expansion gaps in Lie groups.

Abstract

Let $G$ be a connected unimodular group equipped with a (left and hence right) Haar measure $\mu_G$, and suppose $A, B \subseteq G$ are nonempty and compact. An inequality by Kemperman gives us $\mu_G(AB)\geq\min\{\mu_G(A)+\mu_G(B),\mu_G(G)\}.$ Our first result determines the conditions for the equality to hold, providing a complete answer to a question asked by Kemperman in 1964. Our second result characterizes compact and connected $G$, $A$, and $B$ that nearly realize equality, with quantitative bounds having the sharp exponent. This can be seen up-to-constant as a $(3k-4)$-theorem for this setting and confirms the connected case of conjectures by Griesmer and by Tao. As an application, we get a measure expansion gap result for connected compact simple Lie groups. The tools developed in our proof include an analysis of the shape of minimally and nearly minimally expanding pairs of sets, a bridge from this to the properties of a certain pseudometric, and a construction of appropriate continuous group homomorphisms to either $\mathbb{R}$ or $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ from the pseudometric.