Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic

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January 2018, 38(1): 169-186. doi: 10.3934/dcds.2018008

Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic

Sanghoon Kwon ^1, and Seonhee Lim ^2,,

1.	Center for Mathematical Challenges, Korea Institute For Advanced Study, Seoul 02455, Korea
2.	Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea

* Corresponding author: Seonhee Lim

Received September 2016 Revised July 2017 Published September 2017

Fund Project: The second author is supported by Samsung Science and Technology Foundation under Project No. SSTF-BA1601-03 and is an associate member of KIAS.

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 C_α.

Keywords: Field of formal series, effective equidistribution, ergodic theorem, Diophantine approximation.

Mathematics Subject Classification: Primary; 28A33;Secondary:37A15, 22E40.

Citation: Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008

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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