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An application of lattice points counting to shrinking target problems
Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic
1. | Center for Mathematical Challenges, Korea Institute For Advanced Study, Seoul 02455, Korea |
2. | Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea |
For a local field K of formal Laurent series and its ring Z of polynomials, we prove a pointwise equidistribution with an error rate of each H-orbit in SL(d, K)/SL(d, Z) for a certain proper subgroup H of a horospherical group, extending a work of Kleinbock-Shi-Weiss.
We obtain an asymptotic formula for the number of integral solutions to the Diophantine inequalities with weights, generalizing a result of Dodson-Kristensen-Levesley. This result enables us to show pointwise equidistribution for unbounded functions of class Cα.
References:
[1] |
J. Athreya, A. Ghosh and A. Prasad,
Ultrametric logarithm laws Ⅱ, Monatsh Math., 167 (2012), 333-356.
doi: 10.1007/s00605-012-0376-y. |
[2] |
J. Athreya, A. Parrish and J. Tseng,
Ergodic theory and Diophantine approximation for linear forms and translation surfaces and linear forms, Nonlinearity, 29 (2016), 2173-2190.
doi: 10.1088/0951-7715/29/8/2173. |
[3] |
M. Dodson, S. Kristensen and J. Levesley,
A quantitative Khintchine-Groshev type theorem over a field of formal series, Indag. Math. (N.S), 16 (2005), 171-177.
doi: 10.1016/S0019-3577(05)80020-5. |
[4] |
M. Einsiedler, G. Margulis, A. Mohammadi and A. Venkatesh, Effective equidistribution and property (τ), preprint, arXiv: 1503.05884. |
[5] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Annals of Mathematics, 147 (1998), 93-141.
doi: 10.2307/120984. |
[6] |
A. Ghosh,
Metric Diophantine approximation over a local field of positive characteristic, J. Number Theory, 124 (2007), 454-469.
doi: 10.1016/j.jnt.2006.10.009. |
[7] |
D. Kleinbock and G. Margulis, On effective equidistribution of expanding translates of certain orbits in the space of lattices, in Number Theory, Analysis and Geometry, Springer, New York, 2012,385–396. |
[8] |
D. Kleinbock, R. Shi and B. Weiss,
Pointwise equidistribution with an error rate and with respect to unbounded functions, Math. Ann., 367 (2017), 857-879.
doi: 10.1007/s00208-016-1404-3. |
[9] |
D. Kleinbock, R. Shi and G. Tomanov,
s-adic version of Minkowskis geometry of numbers and Mahlers compactness criterion, J. Number Theory, 174 (2017), 150-163.
doi: 10.1016/j.jnt.2016.10.016. |
[10] |
D. Kleinbock and G. Tomanov,
Flows on s-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Coom. Math. Helv., 82 (2007), 519-581.
doi: 10.4171/CMH/102. |
[11] |
A. Mohammadi,
Measures invariant under horospherical subgroups in positive characteristic, J. Mod. Dynamics, 5 (2011), 237-254.
doi: 10.3934/jmd.2011.5.237. |
[12] |
M. Morishita, A mean value theorem in adele geometry, Algebraic number theory and Fermat’s problem, Sūrikaisekikenkyūsho Kkyōroku, (Japanese) (1995), 1-11. |
[13] |
H. Oh,
Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., 113 (2002), 133-192.
doi: 10.1215/S0012-7094-02-11314-3. |
[14] |
M. Rosen, Number Theory in Function Fields, Springer-Verlag, New York, 2002. |
[15] |
R. Rühr, Some Applications of Effective Unipotent Dynamics, Ph. D. Thesis, ETH Zurich, 2015. |
[16] |
N. Shah,
Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. (Math. Sci.), 106 (1996), 105-125.
doi: 10.1007/BF02837164. |
[17] |
R. Shi, Expanding cone and applications to homogeneous dynamics, preprint, arXiv: 1510.05256. |
[18] |
C. Siegel,
Amean value theorem in geometry of numbers, Annals of Mathematics, 46 (1945), 340-347.
doi: 10.2307/1969027. |
[19] |
V. Sprindzuk, Metric Theory of Diophantine Approximations, V. H. Winston & Sons, Washington, DC, 1979. |
[20] |
G. Tomanov,
Orbits on homogeneous spaces of arithmetic origin and approximations, Adv. studies in Pure Math., 26 (2000), 265-297.
|
show all references
References:
[1] |
J. Athreya, A. Ghosh and A. Prasad,
Ultrametric logarithm laws Ⅱ, Monatsh Math., 167 (2012), 333-356.
doi: 10.1007/s00605-012-0376-y. |
[2] |
J. Athreya, A. Parrish and J. Tseng,
Ergodic theory and Diophantine approximation for linear forms and translation surfaces and linear forms, Nonlinearity, 29 (2016), 2173-2190.
doi: 10.1088/0951-7715/29/8/2173. |
[3] |
M. Dodson, S. Kristensen and J. Levesley,
A quantitative Khintchine-Groshev type theorem over a field of formal series, Indag. Math. (N.S), 16 (2005), 171-177.
doi: 10.1016/S0019-3577(05)80020-5. |
[4] |
M. Einsiedler, G. Margulis, A. Mohammadi and A. Venkatesh, Effective equidistribution and property (τ), preprint, arXiv: 1503.05884. |
[5] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Annals of Mathematics, 147 (1998), 93-141.
doi: 10.2307/120984. |
[6] |
A. Ghosh,
Metric Diophantine approximation over a local field of positive characteristic, J. Number Theory, 124 (2007), 454-469.
doi: 10.1016/j.jnt.2006.10.009. |
[7] |
D. Kleinbock and G. Margulis, On effective equidistribution of expanding translates of certain orbits in the space of lattices, in Number Theory, Analysis and Geometry, Springer, New York, 2012,385–396. |
[8] |
D. Kleinbock, R. Shi and B. Weiss,
Pointwise equidistribution with an error rate and with respect to unbounded functions, Math. Ann., 367 (2017), 857-879.
doi: 10.1007/s00208-016-1404-3. |
[9] |
D. Kleinbock, R. Shi and G. Tomanov,
s-adic version of Minkowskis geometry of numbers and Mahlers compactness criterion, J. Number Theory, 174 (2017), 150-163.
doi: 10.1016/j.jnt.2016.10.016. |
[10] |
D. Kleinbock and G. Tomanov,
Flows on s-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Coom. Math. Helv., 82 (2007), 519-581.
doi: 10.4171/CMH/102. |
[11] |
A. Mohammadi,
Measures invariant under horospherical subgroups in positive characteristic, J. Mod. Dynamics, 5 (2011), 237-254.
doi: 10.3934/jmd.2011.5.237. |
[12] |
M. Morishita, A mean value theorem in adele geometry, Algebraic number theory and Fermat’s problem, Sūrikaisekikenkyūsho Kkyōroku, (Japanese) (1995), 1-11. |
[13] |
H. Oh,
Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., 113 (2002), 133-192.
doi: 10.1215/S0012-7094-02-11314-3. |
[14] |
M. Rosen, Number Theory in Function Fields, Springer-Verlag, New York, 2002. |
[15] |
R. Rühr, Some Applications of Effective Unipotent Dynamics, Ph. D. Thesis, ETH Zurich, 2015. |
[16] |
N. Shah,
Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. (Math. Sci.), 106 (1996), 105-125.
doi: 10.1007/BF02837164. |
[17] |
R. Shi, Expanding cone and applications to homogeneous dynamics, preprint, arXiv: 1510.05256. |
[18] |
C. Siegel,
Amean value theorem in geometry of numbers, Annals of Mathematics, 46 (1945), 340-347.
doi: 10.2307/1969027. |
[19] |
V. Sprindzuk, Metric Theory of Diophantine Approximations, V. H. Winston & Sons, Washington, DC, 1979. |
[20] |
G. Tomanov,
Orbits on homogeneous spaces of arithmetic origin and approximations, Adv. studies in Pure Math., 26 (2000), 265-297.
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