# American Institute of Mathematical Sciences

May  2019, 39(5): 2343-2359. doi: 10.3934/dcds.2019099

## On density of infinite subsets I

 Department of Mathematics, University of Maryland College Park, College Park, MD 20742, USA

Received  June 2017 Revised  July 2018 Published  January 2019

Fund Project: Research was partially supported by NSF grant DMS 1602409

Let
 $Y$
be a compact metric space,
 $G$
be a group acting by transformations on
 $Y$
. For any infinite subset
 $A\subset Y$
, we study the density of
 $gA$
for
 $g\in G$
and quantitative density of the set
 ${\bigcup\limits_{g\in G_n}gA}$
by the Hausdorff semimetric
 $d^H$
, for a family of increasing subsets
 $G_n\subset G$
. It is proven that for any integer
 $n\ge 2$
,
 $\epsilon>0$
, any infinite subset
 $A\subset \mathbb T^n$
, there is a
 $g\in SL(n,\mathbb Z)$
such that
 $gA$
is
 $\epsilon$
-dense. We also show that, for any infinite subset
 $A\subset [0,1]$
, for a.e. rotation and a.e. 3-IET,
 $\liminf\limits_nn\cdot d^H\left(\bigcup\limits_{k = 0}^{n-1}T^kA,[0,1]\right) = 0.$
Citation: Changguang Dong. On density of infinite subsets I. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2343-2359. doi: 10.3934/dcds.2019099
##### References:
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##### References:
 [1] N. Alon and Y. Peres, Uniform dilations, Geom. Funct. Anal., 2 (1992), 1-28. doi: 10.1007/BF01895704. Google Scholar [2] Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (Ⅲ), Annals of Mathematics, 178 (2013), 1017-1059. doi: 10.4007/annals.2013.178.3.5. Google Scholar [3] D. Berend and Y. Peres, Asymptotically dense dilations of sets on the circle, J. London Math. Soc., (2) 47 (1993), 1-17. doi: 10.1112/jlms/s2-47.1.1. Google Scholar [4] J. Bourgain, A. Furman, E. Lindenstrauss and S. Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, Journal of the American Mathematical Society, 24 (2011), 231-280. doi: 10.1090/S0894-0347-2010-00674-1. Google Scholar [5] J. Bourgain, E. Lindenstrauss, Ph. Michel and A. Venkatesh, Some effective results for ×a; ×b, Ergodic Theory and Dynamical Systems, 29 (2009), 1705-1722. doi: 10.1017/S0143385708000898. Google Scholar [6] C. Dong, On density of infinite subsets Ⅱ: Dynamics on homogeneous spaces, to appear in Proceedings of the AMS. doi: 10.1090/proc/14298. Google Scholar [7] R. J. Duffin and A. C. Schaeffer, Khintchine's problem in metric Diophantine approximation, Duke Math. J, 8 (1941), 243-255. doi: 10.1215/S0012-7094-41-00818-9. Google Scholar [8] S. Glasner, Almost periodic sets and measures on the torus, Israel J. Math., 32 (1979), 161-172. doi: 10.1007/BF02764912. Google Scholar [9] Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory and Dynamical Systems, 24 (2004), 767-802. doi: 10.1017/S0143385703000440. Google Scholar [10] Peter Humphries, http://mathoverflow.net/a/261553/99556Google Scholar [11] B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity, Annals of Mathematics, 174 (2011), 361-400. doi: 10.4007/annals.2011.174.1.10. Google Scholar [12] H. Kato, Continuum-wise expansive homeomorphisms, Can. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4. Google Scholar [13] A. Katok and A. M. Stepin, Approximation of ergodic dynamical systems by periodic transformations, Soviet Math. Dokl, 7 (1966), 1638-1641. Google Scholar [14] A. Katok, S Katok and K. Schmidt, Rigidity of measurable structure for $\mathbb Z^d$ actions by automorphisms of a torus, Comm. Math. Helv., 77 (2002), 718-745. doi: 10.1007/PL00012439. Google Scholar [15] M. Kelly and T. Lê, Uniform dilations in higher dimensions, J. London Math. Soc., (2) 88 (2013), 925-940. doi: 10.1112/jlms/jdt054. Google Scholar [16] A. I. Khintchine, Continued Fractions, University of Chicago Press, 1964. [17] R. Nair and S. Velani, Glasner sets and polynomials in primes, Proceedings of the American Mathematical Society, 126 (1998), 2835-2840. doi: 10.1090/S0002-9939-98-04396-2. Google Scholar [18] J. Rodriguez Hertz, There are no stable points for continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598, https://arXiv.org/abs/math/0208102v3 doi: 10.4153/CJM-1993-030-4. Google Scholar [19] Z. Wang, Quantitative density under higher rank abelian algebraic toral actions, International Mathematics Research Notices, (2010), 3744-3821. doi: 10.1093/imrn/rnq222. Google Scholar
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