Inhomogeneous Diophantine approximation over fields of formal power series
DOI:
https://doi.org/10.7146/math.scand.a-121462Abstract
We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb {F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline {y}$ in $\mathbb {F}_q((T^{-1}))^2$ by the $\mathrm {SL}_2(\mathbb {F}_q[T])$-orbit of a given point $\underline {x}$ in $\mathbb {F}_q((T^{-1}))^2$.
References
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Bugeaud, Y. and Zhang, Z., On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series, Pacific J. Math. 302 (2019), no. 2, 453–480. https://doi.org/10.2140/pjm.2019.302.453
Cassels, J. W. S., An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957.
Fuchs, M., Metrical theorems for inhomogeneous Diophantine approximation in positive characteristic, Acta Arith. 141 (2010), no. 2, 191–208. https://doi.org/10.4064/aa141-2-4
Ghosh, A., Gorodnik, A., and Nevo, A., Diophantine approximation exponents on homogeneous varieties, in “Recent trends in ergodic theory and dynamical systems”, Contemp. Math., vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 181–200. https://doi.org/10.1090/conm/631/12603
Ghosh, A., Gorodnik, A., and Nevo, A., Best possible rates of distribution of dense lattice orbits in homogeneous spaces, J. Reine Angew. Math. 745 (2018), 155–188. https://doi.org/10.1515/crelle-2016-0001
Khinchin, A. Y., Continued fractions, The University of Chicago Press, Chicago, Ill.-London, 1964.
Kim, D. H. and Nakada, H., Metric inhomogeneous Diophantine approximation on the field of formal Laurent series, Acta Arith. 150 (2011), no. 2, 129–142. https://doi.org/10.4064/aa150-2-3
Kristensen, S., Metric inhomogeneous Diophantine approximation in positive characteristic, Math. Scand. 108 (2011), no. 1, 55–76. https://doi.org/10.7146/math.scand.a-15160
Laurent, M. and Nogueira, A., Approximation to points in the plane by $\mathrm SL(2,\mathbb Z)$-orbits, J. Lond. Math. Soc. (2) 85 (2012), no. 2, 409–429. https://doi.org/10.1112/jlms/jdr061
Ma, C. and Su, W.-Y., Inhomogeneous Diophantine approximation over the field of formal Laurent series, Finite Fields Appl. 14 (2008), no. 2, 361–378. https://doi.org/10.1016/j.ffa.2007.01.004
Maucourant, F. and Weiss, B., Lattice actions on the plane revisited, Geom. Dedicata 157 (2012), 1–21. https://doi.org/10.1007/s10711-011-9596-x
Pollicott, M., Rates of convergence for linear actions of cocompact lattices on the complex plane, Integers 11B (2011), paper no. A12, 7 pp.
Rosen, M., Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. https://doi.org/10.1007/978-1-4757-6046-0
Schmidt, W. M., On continued fractions and Diophantine approximation in power series fields, Acta Arith. 95 (2000), no. 2, 139–166. https://doi.org/10.4064/aa-95-2-139-166
Singhal, L., Diophantine exponents for standard linear actions of $\mathrm SL_2$ over discrete rings in ℂ, Acta Arith. 177 (2017), no. 1, 53–73. https://doi.org/10.4064/aa8370-6-2016
Sprindžuk, V. G., Mahler's problem in metric number theory, Translations of Mathematical Monographs, vol. 25, American Mathematical Society, Providence, R.I., 1969.
Zheng, Z., Simultaneous diophantine approximation in function fields, eprint arXiv:1711.03721 [math.NT], 2017.
Bugeaud, Y. and Zhang, Z., On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series, Pacific J. Math. 302 (2019), no. 2, 453–480. https://doi.org/10.2140/pjm.2019.302.453
Cassels, J. W. S., An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957.
Fuchs, M., Metrical theorems for inhomogeneous Diophantine approximation in positive characteristic, Acta Arith. 141 (2010), no. 2, 191–208. https://doi.org/10.4064/aa141-2-4
Ghosh, A., Gorodnik, A., and Nevo, A., Diophantine approximation exponents on homogeneous varieties, in “Recent trends in ergodic theory and dynamical systems”, Contemp. Math., vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 181–200. https://doi.org/10.1090/conm/631/12603
Ghosh, A., Gorodnik, A., and Nevo, A., Best possible rates of distribution of dense lattice orbits in homogeneous spaces, J. Reine Angew. Math. 745 (2018), 155–188. https://doi.org/10.1515/crelle-2016-0001
Khinchin, A. Y., Continued fractions, The University of Chicago Press, Chicago, Ill.-London, 1964.
Kim, D. H. and Nakada, H., Metric inhomogeneous Diophantine approximation on the field of formal Laurent series, Acta Arith. 150 (2011), no. 2, 129–142. https://doi.org/10.4064/aa150-2-3
Kristensen, S., Metric inhomogeneous Diophantine approximation in positive characteristic, Math. Scand. 108 (2011), no. 1, 55–76. https://doi.org/10.7146/math.scand.a-15160
Laurent, M. and Nogueira, A., Approximation to points in the plane by $\mathrm SL(2,\mathbb Z)$-orbits, J. Lond. Math. Soc. (2) 85 (2012), no. 2, 409–429. https://doi.org/10.1112/jlms/jdr061
Ma, C. and Su, W.-Y., Inhomogeneous Diophantine approximation over the field of formal Laurent series, Finite Fields Appl. 14 (2008), no. 2, 361–378. https://doi.org/10.1016/j.ffa.2007.01.004
Maucourant, F. and Weiss, B., Lattice actions on the plane revisited, Geom. Dedicata 157 (2012), 1–21. https://doi.org/10.1007/s10711-011-9596-x
Pollicott, M., Rates of convergence for linear actions of cocompact lattices on the complex plane, Integers 11B (2011), paper no. A12, 7 pp.
Rosen, M., Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. https://doi.org/10.1007/978-1-4757-6046-0
Schmidt, W. M., On continued fractions and Diophantine approximation in power series fields, Acta Arith. 95 (2000), no. 2, 139–166. https://doi.org/10.4064/aa-95-2-139-166
Singhal, L., Diophantine exponents for standard linear actions of $\mathrm SL_2$ over discrete rings in ℂ, Acta Arith. 177 (2017), no. 1, 53–73. https://doi.org/10.4064/aa8370-6-2016
Sprindžuk, V. G., Mahler's problem in metric number theory, Translations of Mathematical Monographs, vol. 25, American Mathematical Society, Providence, R.I., 1969.
Zheng, Z., Simultaneous diophantine approximation in function fields, eprint arXiv:1711.03721 [math.NT], 2017.
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Published
2020-09-03
How to Cite
Bugeaud, Y., Singhal, L., & Zhang, Z. (2020). Inhomogeneous Diophantine approximation over fields of formal power series. MATHEMATICA SCANDINAVICA, 126(3), 451–478. https://doi.org/10.7146/math.scand.a-121462
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