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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Subsets close to invariant subsets for group actions


Authors: Leonid Brailovsky, Dmitrii V. Pasechnik and Cheryl E. Praeger
Journal: Proc. Amer. Math. Soc. 123 (1995), 2283-2295
MSC: Primary 20B05; Secondary 20B07
MathSciNet review: 1307498
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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 $ \vert{A^g}\backslash A\vert \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 $ \vert A\vartriangle \Gamma \vert < 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 $ \vert A\vartriangle \Gamma \vert$, in terms of both m and k, are also obtained.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1307498-3
PII: S 0002-9939(1995)1307498-3
Article copyright: © Copyright 1995 American Mathematical Society