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  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

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