Subsets close to invariant subsets for group actions
Leonid Brailovsky, Dmitrii V. Ṗasechnik, Cheryl E. Praeger · Proceedings of the American Mathematical Society · 1995
Let G be a group acting on a set Ω \Omega and k a non-negative integer. A subset (finite or infinite) A ⊆ Ω A \subseteq \Omega is called k -quasi-invariant if | A g ∖ A | ≤ k |{A^g}\backslash A| \leq k for every g ∈ G g \in G . It is shown that if A is k -quasi-invariant for k ≥ 1 k \geq 1 , then there exists an invariant subset Γ ⊆ Ω \Gamma \subseteq \Omega such that | A △ Γ | > 2 e k ⌈ ( ln 2 k ) ⌉ |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 2 k − 1 2k - 1 . Certain other bounds on