American Institute of Mathematical Sciences

Advanced
  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]

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
[7]

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
[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]

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
[16]

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
[17]

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
[18]

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492
[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

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (31)
  • HTML views (70)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences