Abstract
Approximate invariance for ergodic actions of amenable groups, Discrete Analysis 2019:6, 56 pp. A basic phenomenon in additive combinatorics is that the 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 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 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 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 Folner 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.
Highlights
With this paper we wish to take the first steps towards what could be called approximate dynamics or approximate ergodic theory, a line of research concerned with the interplay between expansion and approximate invariance of so called action sets, dynamical analogues of product sets in groups
Our second main theorem asserts that upon passing to factors, the setting described above is the only source of examples of ergodic Borel G-spaces for which the lower bound in Theorem 1.6 is attained
Our inspiration for this subsection comes from a classical result of Kneser [21], generalizing an earlier landmark made by Mann [24]. It is an inverse result for subsets A, B ⊂ N with positive lower asymptotic densities along the Følner sequence ([1, n]) such that d[1,n](A + B) < min(1, d[1,n](A) + d[1,n](B)), and roughly asserts that A and B must be contained in periodic subsets which are not "much larger" in size than A and B, and A + B is a co-finite subset of a periodic set in N
Summary
Subsets of locally compact groups which are almost closed under multiplication are classical objects of study in harmonic analysis, number theory and geometry, and continually appear in new applications. Let us begin by making a trivial observation: If G Y, k-doubling sets in G naturally give rise to k-approximately invariant subsets in Y as follows. As a warm-up, we provide a classification of 2-doubling pairs (A, B), where A is a "large" and "aperiodic" (or "spread-out") subset of a countable (infinite) abelian group G, for instance (Z, +), and B is a Borel set with positive measure in some ergodic Borel G-space (Y, ν). (ii) a G-factor map σ : (Y, ν) → (T, mT) and a closed interval Jo ⊂ T with mT(Jo) = ν(B), where the G-action on T is defined as in (1.1) using the group compactification (T, τ), such that A ⊂ τ−1(Io) and B = σ−1(Jo) modulo ν-null sets. We stress that the converse holds; if A and B are as in the conclusion of Theorem 1.5, (A, B) is 2-doubling (modulo null sets)
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