Approximation of points in the plane by generic lattice orbits

Previous Article
Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups
JMD Home
This Volume
Next Article
On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$)

2017, 11: 143-153. doi: 10.3934/jmd.2017007

Approximation of points in the plane by generic lattice orbits

Dubi Kelmer

Department of Mathematics, Maloney Hall, Boston College, Chestnut Hill, MA 02467-3806, USA

Received June 29, 2016 Revised December 04, 2016 Published February 2017

Fund Project: Partially supported by NSF grant DMS-1401747

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma < {\rm{S}}{{\rm{L}}_2}\left( {\mathbb{R}} \right)$ acting linearly on ${\mathbb{R}^2}$. Our method gives bounds that are uniform for almost all orbits.

Keywords: Diophantine approximation, lattice action, shrinking targets.

Mathematics Subject Classification: Primary: 11J20; Secondary: 37A17.

Citation: Dubi Kelmer. Approximation of points in the plane by generic lattice orbits. Journal of Modern Dynamics, 2017, 11: 143-153. doi: 10.3934/jmd.2017007

References:

[1]	A. Ghosh, A. Gorodnik and A. Nevo, Best possible rates of distribution of dense lattice orbits in homogeneous spaces, J. Reine Angew. Math., to appear. doi: 10.1515/crelle-2016-0001. Google Scholar
[2]	A. Ghosh, A. Gorodnik and A. Nevo, Diophantine approximation exponents on homogeneous varieties, in Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, (2015), 181-200. doi: 10.1090/conm/631/12603. Google Scholar
[3]	A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, arXiv: 1512.05848, 2015.Google Scholar
[4]	A. Gorodnik and B. Weiss, Distribution of lattice orbits on homogeneous varieties, Geom. Funct. Anal., 17 (2007), 58-115. doi: 10.1007/s00039-006-0583-6. Google Scholar
[5]	D. Kelmer, Shrinking targets for discrete time flows on hyperbolic manifolds, preprint.Google Scholar
[6]	H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc., 16 (2003), 139-183. doi: 10.1090/S0894-0347-02-00410-1. Google Scholar
[7]	F. Ledrappier, Distribution des orbites des réseaux sur le plan réel, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 61-64. doi: 10.1016/S0764-4442(99)80462-5. Google Scholar
[8]	M. Laurent and A. Nogueira, Approximation to points in the plane by SL(2.Z)-orbits, J. Lond. Math. Soc. (2), 85 (2012), 409-429. doi: 10.1112/jlms/jdr061. Google Scholar
[9]	M. Laurent and A. Nogueira, Inhomogeneous approximation with coprime integers and lattice orbits, Acta Arith., 154 (2012), 413-427. doi: 10.4064/aa154-4-5. Google Scholar
[10]	F. Maucourant and B. Weiss, Lattice actions on the plane revisited, Geom. Dedicata, 157 (2012), 1-21. doi: 10.1007/s10711-011-9596-x. Google Scholar
[11]	A. Nogueira, Orbit distribution on R2 under the natural action of SL(2.Z), Indag. Math. (N.S.), Indag. Math. (N.S.), 13 (2002), 103-124. doi: 10.1016/S0019-3577(02)90009-1. Google Scholar
[12]	M. Pollicott, Rates of convergence for linear actions of cocompact lattices on the complex plane, Integers, 11B (2011), Paper No. A12, 7pp. Google Scholar
[13]	L. Singhal, Diophantine exponents for standard linear actions of SL_2 over discrete rings in C, Acta Arith., 177 (2017), 53-73. doi: 10.4064/aa8370-6-2016. Google Scholar
[14]	A. Venkatesh, Sparse equidistribution problems. period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094. doi: 10.4007/annals.2010.172.989. Google Scholar

show all references

References:

[1]	A. Ghosh, A. Gorodnik and A. Nevo, Best possible rates of distribution of dense lattice orbits in homogeneous spaces, J. Reine Angew. Math., to appear. doi: 10.1515/crelle-2016-0001. Google Scholar
[2]	A. Ghosh, A. Gorodnik and A. Nevo, Diophantine approximation exponents on homogeneous varieties, in Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, (2015), 181-200. doi: 10.1090/conm/631/12603. Google Scholar
[3]	A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, arXiv: 1512.05848, 2015.Google Scholar
[4]	A. Gorodnik and B. Weiss, Distribution of lattice orbits on homogeneous varieties, Geom. Funct. Anal., 17 (2007), 58-115. doi: 10.1007/s00039-006-0583-6. Google Scholar
[5]	D. Kelmer, Shrinking targets for discrete time flows on hyperbolic manifolds, preprint.Google Scholar
[6]	H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc., 16 (2003), 139-183. doi: 10.1090/S0894-0347-02-00410-1. Google Scholar
[7]	F. Ledrappier, Distribution des orbites des réseaux sur le plan réel, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 61-64. doi: 10.1016/S0764-4442(99)80462-5. Google Scholar
[8]	M. Laurent and A. Nogueira, Approximation to points in the plane by SL(2.Z)-orbits, J. Lond. Math. Soc. (2), 85 (2012), 409-429. doi: 10.1112/jlms/jdr061. Google Scholar
[9]	M. Laurent and A. Nogueira, Inhomogeneous approximation with coprime integers and lattice orbits, Acta Arith., 154 (2012), 413-427. doi: 10.4064/aa154-4-5. Google Scholar
[10]	F. Maucourant and B. Weiss, Lattice actions on the plane revisited, Geom. Dedicata, 157 (2012), 1-21. doi: 10.1007/s10711-011-9596-x. Google Scholar
[11]	A. Nogueira, Orbit distribution on R2 under the natural action of SL(2.Z), Indag. Math. (N.S.), Indag. Math. (N.S.), 13 (2002), 103-124. doi: 10.1016/S0019-3577(02)90009-1. Google Scholar
[12]	M. Pollicott, Rates of convergence for linear actions of cocompact lattices on the complex plane, Integers, 11B (2011), Paper No. A12, 7pp. Google Scholar
[13]	L. Singhal, Diophantine exponents for standard linear actions of SL_2 over discrete rings in C, Acta Arith., 177 (2017), 53-73. doi: 10.4064/aa8370-6-2016. Google Scholar
[14]	A. Venkatesh, Sparse equidistribution problems. period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094. doi: 10.4007/annals.2010.172.989. Google Scholar

[1]	Dmitry Kleinbock, Xi Zhao. An application of lattice points counting to shrinking target problems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 155-168. doi: 10.3934/dcds.2018007
[2]	Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129
[3]	Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[4]	Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104
[5]	Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008
[6]	Jimmy Tseng. On circle rotations and the shrinking target properties. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1111-1122. doi: 10.3934/dcds.2008.20.1111
[7]	Jörg Schmeling. A notion of independence via moving targets. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 269-280. doi: 10.3934/dcds.2006.15.269
[8]	Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251
[9]	Sara D. Cardell, Amparo Fúster-Sabater. Modelling the shrinking generator in terms of linear CA. Advances in Mathematics of Communications, 2016, 10 (4) : 797-809. doi: 10.3934/amc.2016041
[10]	Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229
[11]	Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511
[12]	David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317
[13]	Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477
[14]	E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
[15]	Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492
[16]	Markku Lehtinen, Baylie Damtie, Petteri Piiroinen, Mikko Orispää. Perfect and almost perfect pulse compression codes for range spread radar targets. Inverse Problems & Imaging, 2009, 3 (3) : 465-486. doi: 10.3934/ipi.2009.3.465
[17]	Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo, Michele Piana. Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering. Inverse Problems & Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231
[18]	Rong Liu, Saini Jonathan Tishari. Automatic tracking and positioning algorithm for moving targets in complex environment. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1251-1264. doi: 10.3934/dcdss.2019086
[19]	S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.
[20]	Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549

American Institute of Mathematical Sciences