Measure doubling in unimodular locally compact groups and quotients

TL;DR

This paper investigates measure doubling properties in unimodular locally compact groups and their quotients, establishing bounds on measure growth under quotient maps and improving previous results.

Contribution

It proves new bounds on measure doubling in quotients of unimodular groups, including sharp constants and a method to find large subsets with controlled measure growth.

Findings

Established that ^2  in quotients is bounded by K^2 times , with sharpness.

Improved previous bounds from K^3 to K^2 under symmetry assumptions.

Identified large subsets with controlled measure doubling in the quotient.

Abstract

We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $\mu_G$, and a compact $A\subseteq G$ of positive measure with $\mu_G(A^2)\leq K\mu_G(A)$. Let $H$ be a closed normal subgroup of G and $\pi: G \rightarrow G/H$ be the quotient map. With the further assumption that $A= A^{-1}$, we show $$\mu_{G/H}(\pi A ^2) \leq K^2 \mu_{G/H}(\pi A).$$ We also demonstrate that $K^2$ cannot be replaced by $(1-\epsilon)K^2$ for any $\epsilon>0$. In the general case (without $A=A^{-1}$), we show $\mu_{G/H}(\pi A ^2) \leq K^3 \mu_{G/H}(\pi A)$, improving an earlier result by An, Jing, Zhang, and the third author. Moreover, we are able to extract a compact set $B\subseteq A$ with $\mu_G(B)> \mu_G(A)/2$ such that $ \mu_{G/H}(\pi B^2) < 2K \mu_{G/H}(\pi B)$.