Subsets close to invariant subsets for group actions
Pasechnik, Dmitrii V.
Praeger, Cheryl E.
Date of Issue1995
School of Physical and Mathematical Sciences
Let G be a group acting on a set Ω and k a non-negative integer. A subset (finite or infinite) A ⊆ Ω is called k-quasi-invariant if |Ag \ A| ≤k for every g ∈ G. It is shown that if A is k-quasi-invariant for k ≥1 , then there exists an invariant subset Γ⊆Ω such that |A Δ Γ | < 2ek [(In 2k)]. 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 Δ Γ |, in terms of both m and k , are also obtained.
Proceedings of the American Mathematical Society
© 1995 American Mathematical Society. This paper was published in Proceedings of the American Mathematical Society and is made available as an electronic reprint (preprint) with permission of American Mathematical Society. The paper can be found at :[DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1307498-3].One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.