Subsets close to invariant subsets for group actions

Abstract:

Let G be a group acting on a set $\Omega$ and k a non-negative integer. A subset (finite or infinite) $A \subseteq \Omega$ is called k-quasi-invariant if $|{A^g}\backslash A| \leq k$ for every $g \in G$. It is shown that if A is k-quasi-invariant for $k \geq 1$, then there exists an invariant subset $\Gamma \subseteq \Omega$ such that $|A\vartriangle \Gamma | < 2ek\left \lceil {(\ln 2k)} \right \rceil$. Information about G-orbit intersections with A is obtained. In particular, the number m of G-orbits which have non-empty intersection with A, but are not contained in A, is at most $2k - 1$. Certain other bounds on $|A\vartriangle \Gamma |$, in terms of both m and k, are also obtained.