INVERSE THEOREMS FOR SETS AND MEASURES OF POLYNOMIAL GROWTH

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 ‘symmetrized’ variant of a recent non-abelian 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.