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On density of infinite subsets I
Department of Mathematics, University of Maryland College Park, College Park, MD 20742, USA |
$ Y $ |
$ G $ |
$ Y $ |
$ A\subset Y $ |
$ gA $ |
$ g\in G $ |
$ {\bigcup\limits_{g\in G_n}gA} $ |
$ d^H $ |
$ G_n\subset G $ |
$ n\ge 2 $ |
$ \epsilon>0 $ |
$ A\subset \mathbb T^n $ |
$ g\in SL(n,\mathbb Z) $ |
$ gA $ |
$ \epsilon $ |
$ A\subset [0,1] $ |
$ \liminf\limits_nn\cdot d^H\left(\bigcup\limits_{k = 0}^{n-1}T^kA,[0,1]\right) = 0. $ |
References:
[1] |
N. Alon and Y. Peres,
Uniform dilations, Geom. Funct. Anal., 2 (1992), 1-28.
doi: 10.1007/BF01895704. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
S. Glasner,
Almost periodic sets and measures on the torus, Israel J. Math., 32 (1979), 161-172.
doi: 10.1007/BF02764912. |
[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. |
[10] |
Peter Humphries, http://mathoverflow.net/a/261553/99556 |
[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. |
[12] |
H. Kato,
Continuum-wise expansive homeomorphisms, Can. J. Math., 45 (1993), 576-598.
doi: 10.4153/CJM-1993-030-4. |
[13] |
A. Katok and A. M. Stepin,
Approximation of ergodic dynamical systems by periodic transformations, Soviet Math. Dokl, 7 (1966), 1638-1641.
|
[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. |
[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. |
[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. |
[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. |
[19] |
Z. Wang, Quantitative density under higher rank abelian algebraic toral actions, International Mathematics Research Notices, (2010), 3744-3821.
doi: 10.1093/imrn/rnq222. |
show all references
References:
[1] |
N. Alon and Y. Peres,
Uniform dilations, Geom. Funct. Anal., 2 (1992), 1-28.
doi: 10.1007/BF01895704. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
S. Glasner,
Almost periodic sets and measures on the torus, Israel J. Math., 32 (1979), 161-172.
doi: 10.1007/BF02764912. |
[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. |
[10] |
Peter Humphries, http://mathoverflow.net/a/261553/99556 |
[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. |
[12] |
H. Kato,
Continuum-wise expansive homeomorphisms, Can. J. Math., 45 (1993), 576-598.
doi: 10.4153/CJM-1993-030-4. |
[13] |
A. Katok and A. M. Stepin,
Approximation of ergodic dynamical systems by periodic transformations, Soviet Math. Dokl, 7 (1966), 1638-1641.
|
[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. |
[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. |
[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. |
[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. |
[19] |
Z. Wang, Quantitative density under higher rank abelian algebraic toral actions, International Mathematics Research Notices, (2010), 3744-3821.
doi: 10.1093/imrn/rnq222. |
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