Inverse theorems for sets and measures of polynomial growth

TL;DR

This paper characterizes finite sets and measures of polynomial growth in groups, showing they are controlled by coset nilprogressions and extending inverse theorems to nonabelian settings.

Contribution

It provides a structural description of polynomial growth sets and measures, generalizing previous inverse theorems and explicitly describing growth behavior.

Findings

Sets of polynomial growth are controlled by coset nilprogressions.

Explicit growth descriptions for powers of sets and measures.

Connections to inverse Littlewood-Offord theorems in abelian and nonabelian cases.

Abstract

We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$, showing that such sets are controlled by (a bounded number of translates of) a coset nilprogression in a certain precise sense. This description recovers some previous results of Breuillard-Green-Tao and Breuillard-Tointon concerning sets of polynomial growth; we are also able to describe the subsequent growth of $|A^m|$ fairly explicitly for $m \geq n$, at least when $A$ is a symmetric neighbourhood of the identity. We also obtain an analogous description of symmetric probability measures $\mu$ whose $n$-fold convolutions $\mu^{*n}$ obey the condition $\| \mu^{*n} \|_{\ell^2}^{-2} \leq n^d \|\mu \|_{\ell^2}^{-2}$. In the abelian case, this description recovers the inverse Littlewood-Offord theorem of Nguyen-Vu, and gives a variant of a recent nonabelian inverse Littlewood-Offord theorem of Tiep-Vu. Our main tool to establish these results is the inverse theorem of Breuillard, Green, and the author that describes the structure of approximate groups.