Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

Abstract

In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

Keywords

Multidimensional multiplicative large sets / Return times / Decompositions of minimal systems / Minimal systems

Funding

NNSF of China(11431012)