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On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$)
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Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups
Approximation of points in the plane by generic lattice orbits
Department of Mathematics, Maloney Hall, Boston College, Chestnut Hill, MA 02467-3806, USA |
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.
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. |
| [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. |
| [3] |
A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, arXiv: 1512.05848, 2015. |
| [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. |
| [5] |
D. Kelmer, Shrinking targets for discrete time flows on hyperbolic manifolds, preprint. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
M. Pollicott,
Rates of convergence for linear actions of cocompact lattices on the complex plane, Integers, 11B (2011), Paper No. A12, 7pp.
|
| [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. |
| [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. |
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. |
| [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. |
| [3] |
A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, arXiv: 1512.05848, 2015. |
| [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. |
| [5] |
D. Kelmer, Shrinking targets for discrete time flows on hyperbolic manifolds, preprint. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
M. Pollicott,
Rates of convergence for linear actions of cocompact lattices on the complex plane, Integers, 11B (2011), Paper No. A12, 7pp.
|
| [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. |
| [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. |
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