Approximate invariance for ergodic actions of amenable groups

Approximate invariance for ergodic actions of amenable groups, Discrete Analysis 2019:6, 56 pp.

A basic phenomenon in additive combinatorics is that the “size” of a sumset $A+B$ or product set $AB$ of two sets $A,B$ is “usually” at least as large as the sum of the “sizes” of the individual sets $A,B$ (or the size of the ambient group $G$, whichever is smaller), unless the sets $A,B$ have an unusually large amount of “periodic” structure; nowadays such results are often referred to as “inverse theorems” for the sum set or product set operation. For instance, a theorem of Kemperman asserts that if $A,B$ are Borel measurable subsets of a compact group $G$ with positive measure with respect to the probability Haar measure $\mu$, then one has $\mu(AB) \geq \min( 1, \mu(A)+\mu(B))$ unless $A$ and $B$ are “periodic” sets, which means that (modulo null sets) they are unions of cosets of an open normal subgroup of $G$. In a similar spirit, a theorem of Kneser asserts that if $A,B$ are sets of natural numbers, and the lower density $\underline{d}(A)$ of a set $A$ is defined as

$$\underline{d}(A) := \liminf_{n \to \infty} \frac{1}{n} |A \cap \{1,\dots,n\}|,$$

then one has $\underline{d}(A+B) \geq \min(1, \underline{d}(A) + \underline{d}(B) )$ unless $A+B$ is a periodic set (modulo a finite set), and also $A$ and $B$ are contained in periodic sets that are not much “larger” than $A,B$ in some sense.

This paper systematically studies these sorts of problems in several rather general, but interrelated, contexts. One of these is the study of lower densities of product sets $AB$ in an arbitrary countable amenable group $G$, in the spirit of Kneser’s theorem. Another is the study of the action of positive lower density subsets $A$ of a countable amenable group $G$ acting on a positive measure subset $B$ of a probability space $X$. The third is the study of product sets on a compact group $G$, in the spirit of Kemperman’s theorem. One of the technical achievements of the paper is to link these three contexts together, using variants of the celebrated Furstenberg correspondence principle, as well as some representation theory.

The precise results of the paper are somewhat technical to state, but one such result of Kneser type is as follows: if $A,B$ are subsets of a countable amenable group with $B$ syndetic, and lower density is taken with respect to an appropriate Følner sequence, then one has $\underline{d}(AB) \geq \underline{d}(A) + \underline{d}(B)$ unless either $A$ is “not spread out” in the sense that it is (modulo sets of zero density) contained in a proper union of cosets of a finite index subgroup, or else $AB$ is “thick” in the sense that it contains translates of any given finite set. There are also more refined results studying when equality can occur in inequalities of the above type.