scholarly journals Metric discrepancy results for geometric progressions perturbed by irrational rotations

2019 ◽  
Vol 161 (1) ◽  
pp. 48-65
Author(s):  
K. Fukuyama ◽  
S. Mori ◽  
Y. Tanabe
Keyword(s):  
Irrational Rotations ◽  
Geometric Progressions

REFLEXIVITY INDEX AND IRRATIONAL ROTATIONS

2021 ◽  
pp. 1-13
Author(s):  
BINGZHANG MA ◽  
K. J. HARRISON
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Irrational Rotations ◽  
Index 2

Abstract We determine the reflexivity index of some closed set lattices by constructing maps relative to irrational rotations. For example, various nests of closed balls and some topological spaces, such as even-dimensional spheres and a wedge of two circles, have reflexivity index 2. We also show that a connected double of spheres has reflexivity index at most 2.

scholarly journals Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

2021 ◽  
Author(s):  
Jiahao Qiu ◽  
Jianjie Zhao
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Minimal System ◽  
Combinatorial Properties ◽  
Return Times ◽  
Residual Set ◽  
Closed Sets ◽  
Open Set ◽  
Geometric Progressions ◽  
Set Of Points

AbstractIn 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.

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