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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 Cα.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Einsiedler, G. Margulis, A. Mohammadi and A. Venkatesh, Effective equidistribution and property (τ), preprint, arXiv: 1503.05884. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Morishita, A mean value theorem in adele geometry, Algebraic number theory and Fermat’s problem, Sūrikaisekikenkyūsho Kkyōroku, (Japanese) (1995), 1-11. Google Scholar

[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.  Google Scholar

[14]

M. Rosen, Number Theory in Function Fields, Springer-Verlag, New York, 2002. Google Scholar

[15]

R. Rühr, Some Applications of Effective Unipotent Dynamics, Ph. D. Thesis, ETH Zurich, 2015. Google Scholar

[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.  Google Scholar

[17]

R. Shi, Expanding cone and applications to homogeneous dynamics, preprint, arXiv: 1510.05256. Google Scholar

[18]

C. Siegel, Amean value theorem in geometry of numbers, Annals of Mathematics, 46 (1945), 340-347.  doi: 10.2307/1969027.  Google Scholar

[19]

V. Sprindzuk, Metric Theory of Diophantine Approximations, V. H. Winston & Sons, Washington, DC, 1979. Google Scholar

[20]

G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, Adv. studies in Pure Math., 26 (2000), 265-297.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Einsiedler, G. Margulis, A. Mohammadi and A. Venkatesh, Effective equidistribution and property (τ), preprint, arXiv: 1503.05884. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Morishita, A mean value theorem in adele geometry, Algebraic number theory and Fermat’s problem, Sūrikaisekikenkyūsho Kkyōroku, (Japanese) (1995), 1-11. Google Scholar

[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.  Google Scholar

[14]

M. Rosen, Number Theory in Function Fields, Springer-Verlag, New York, 2002. Google Scholar

[15]

R. Rühr, Some Applications of Effective Unipotent Dynamics, Ph. D. Thesis, ETH Zurich, 2015. Google Scholar

[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.  Google Scholar

[17]

R. Shi, Expanding cone and applications to homogeneous dynamics, preprint, arXiv: 1510.05256. Google Scholar

[18]

C. Siegel, Amean value theorem in geometry of numbers, Annals of Mathematics, 46 (1945), 340-347.  doi: 10.2307/1969027.  Google Scholar

[19]

V. Sprindzuk, Metric Theory of Diophantine Approximations, V. H. Winston & Sons, Washington, DC, 1979. Google Scholar

[20]

G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, Adv. studies in Pure Math., 26 (2000), 265-297.   Google Scholar

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