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:
[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. BourgainA. FurmanE. 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. BourgainE. LindenstraussPh. 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/261553/99556 Google Scholar

[11]

B. KalininA. 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. KatokS 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.   Google Scholar
[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

show all references

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. BourgainA. FurmanE. 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. BourgainE. LindenstraussPh. 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/261553/99556 Google Scholar

[11]

B. KalininA. 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. KatokS 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.   Google Scholar
[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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