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Simultaneous p-adic Diophantine approximation

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  31 January 2023

V. BERESNEVICH ,
J. LEVESLEY  and
B. WARD
V. BERESNEVICH
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: [email protected], [email protected], [email protected]
J. LEVESLEY
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: [email protected], [email protected], [email protected]
B. WARD
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: [email protected], [email protected], [email protected]
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Abstract

The aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approximation in the p-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on p-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu in 2019, and a Zero-One law for weighted p-adic approximations established in this paper.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11J61: Approximation in non-Archimedean valuations
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 175 , Issue 1 , July 2023 , pp. 13 - 50
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Allen, D. and Baker, S.. A general mass transference principle. Selecta Math. (N.S.), 25(3) (2019).CrossRefGoogle Scholar
Allen, D. and Beresnevich, V.. A mass transference principle for systems of linear forms and its applications. Compositio Math. 154(5) (2018), 1014–1047.CrossRefGoogle Scholar
Allen, D. and Troscheit, S.. The mass transference principle: ten years on. In Horizons of fractal geometry and complex dimensions. Contemp. Math. vol. 731 (Amer. Math. Soc., Providence, RI, 2019), pp. 133.CrossRefGoogle Scholar
Badziahin, D. and Bugeaud, Y.. On simultaneous rational approximation to a real number and its integral powers, II. New York J. Math. 26 (2020), 362–377.Google Scholar
Badziahin, D. and Levesley, J.. A note on simultaneous and multiplicative Diophantine approximation on planar curves. Glasg. Math. J. 49(2) (2007), 367–375.CrossRefGoogle Scholar
Badziahin, D., Bugeaud, Y. and Schleischitz, J.. On simultaneous rational approximation to a p-adic number and its integral powers, II. Proc. Edinburgh Math. Soc. (2), 64(2) (2021), 317–337.CrossRefGoogle Scholar
Beresnevich, V.. Rational points near manifolds and metric Diophantine approximation. Ann. of Math. (2), 175(1) (2012), 187–235.CrossRefGoogle Scholar
Beresnevich, V., Bernik, V. and Kovalevskaya, E.. On approximation of p-adic numbers by p-adic algebraic numbers. J. Number Theory 111(1) (2005), 33–56.CrossRefGoogle Scholar
Beresnevich, V., Dickinson, D. and Velani, S.. Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179(846) (2006), x+91.Google Scholar
Beresnevich, V., Dickinson, D. and Velani, S.. Diophantine approximation on planar curves and the distribution of rational points. Ann. of Math. (2), 166(2) (2007), 367–426. With an Appendix II by R. C. Vaughan.CrossRefGoogle Scholar
Beresnevich, V. and Kovalevskaya, È.. On Diophantine approximations of dependent quantities in the p-adic case. Mat. Zametki 73(1) (2003), 22–37.Google Scholar
Beresnevich, V., Lee, L., Vaughan, R.C. and Velani, S.. Diophantine approximation on manifolds and lower bounds for Hausdorff dimension. Mathematika 63(3) (2017), 762–779.CrossRefGoogle Scholar
Beresnevich, V., Levesley, J. and Ward, B.. A lower bound for the Hausdorff dimension of the set of weighted simultaneously approximable points over manifolds. Int. J. Number Theory 17(8) (2021), 1795–1814.CrossRefGoogle Scholar
Beresnevich, V., Ramrez, F. and Velani, S.. Metric Diophantine approximation: aspects of recent work. In Dynamics and Analytic Number Theory London Math. Soc. Lecture Note Ser. vol. 437 (Cambridge University Press, Cambridge, 2016), pp. 1–95.Google Scholar
Beresnevich, V., Vaughan, R. C., Velani, S. and Zorin, E.. Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory. Int. Math. Res. Not. IMRN 2017(10) (2017), 2885–2908.Google Scholar
Beresnevich, V., Vaughan, R. C., Velani, S. and Zorin, E.. Diophantine approximation on curves and the distribution of rational points: contributions to the divergence theory. Adv. Math. 388(33) (2021).CrossRefGoogle Scholar
Beresnevich, V., Vaughan, R.C. and Velani, S.. Inhomogeneous Diophantine approximation on planar curves. Math. Ann. 349(4) (2011), 929–942.CrossRefGoogle Scholar
Beresnevich, V. and Velani, S. A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2), 164(3) (2006), 971–992.CrossRefGoogle Scholar
Beresnevich, V. and Velani, S.. Schmidt’s theorem, Hausdorff measures, and slicing. Int. Math. Res. Not. 24 (2006) art id. 48794.Google Scholar
Beresnevich, V. and Velani, S.. A note on simultaneous Diophantine approximation on planar curves. Math. Ann. 337(4) (2007), 769–796.CrossRefGoogle Scholar
Beresnevich, V. and Velani, S.. A note on zero-one laws in metrical Diophantine approximation. Acta Arith. 133(4) (2008), 363–374.CrossRefGoogle Scholar
Beresnevich, V. and Yang, L.. Khintchine’s theorem and diophantine approximation on manifolds. https://arxiv.org/abs/2105.13872.Google Scholar
Beresnevich, V. and Zorin, E.. Explicit bounds for rational points near planar curves and metric Diophantine approximation. Adv. Math. 225(6) (2010), 3064–3087.CrossRefGoogle Scholar
Bernik, V. and Dodson, M.. Metric Diophantine Approximation on Manifolds. Cambridge Tracts in Math. vol. 137 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
Besicovitch, A.. Sets of fractional dimensions (IV): On rational approximation to real numbers. J. London Math. Soc. 9(2) (1934), 126–131.CrossRefGoogle Scholar
Billingsley, P.. Probability and measure. Wiley Series in Probability and Mathematical Statistics (John Wiley and Sons Inc., New York, third edition, 1995). A Wiley-Interscience Publication.Google Scholar
Borel, M. E.. Les probabilites d’enombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo (1884-1940), 27(1) (1909), 247–271.CrossRefGoogle Scholar
Budarina, N.. Diophantine approximation on the curves with non-monotonic error function in the p-adic case. Chebyshevski Sb. 11(1(33)) (2010), 74–80.Google Scholar
Budarina, N.. Simultaneous Diophantine approximation in the real and p-adic fields with nonmonotonic error function. Lithuanian Math. J. 51(4) (2011), 461–471.CrossRefGoogle Scholar
Budarina, N., Dickinson, D. and Bernik, V.. Simultaneous Diophantine approximation in the real, complex and p-adic fields. Math. Proc. Camb. Phil. Soc. 149(2) (2010), 193–216.CrossRefGoogle Scholar
Budarina, N., Dickinson, D. and Levesley, J.. Simultaneous Diophantine approximation on polynomial curves. Mathematika 56(1) (2010), 77–85.CrossRefGoogle Scholar
Bugeaud, Y., Budarina, N., Dickinson, D. and O’Donnell, H.. On simultaneous rational approximation to a p-adic number and its integral powers. Proc. Edinburgh Math. Soc. (2), 54(3) (2011), 599–612.CrossRefGoogle Scholar
Datta, S. and Ghosh, A.. Multiplicative p-adic metric Diophantine approximation on manifolds and dichotomy of exponents. Bull. Soc. Math. France 148(4) (2020), 733–747.CrossRefGoogle Scholar
Datta, S. and Ghosh, A.. Diophantine inheritance for p-adic measures. Ann. Scuola Norm. Super. Pisa Cl. Sci. (5), 23(1) (2022), 49–80.Google Scholar
Duffin, R. and Schaeffer, A.. Khintchine’s problem in metric Diophantine approximation. Duke Math. J. 8 (1941), 243–255.CrossRefGoogle Scholar
Erdös, P. and Rényi, A.. On Cantor’s series with convergent $\sum 1/q_{n}$ . Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 2 (1959), 93–109.Google Scholar
Falconer, K.. Fractal Geometry. (John Wiley and Sons, Ltd., Chichester, third edition, 2014). Mathematical foundations and applications.Google Scholar
Federer, H.. Geometric measure theory. Grundlehren Math. Wiss. Band 153 (Springer-Verlag New York Inc., New York, 1969).Google Scholar
Gouvêa, F.. Elementary analysis in $\mathbb{Q}_{p}$ . In p-adic Numbers, (Springer, 2020), pp. 109–165.CrossRefGoogle Scholar
Harman, G.. Metric Number Theory London Math. Soc. Monogr. N.S. vol. 18 (Clarendon Press, Oxford University Press, New York, 1998).Google Scholar
Haynes, A.. The metric theory of p-adic approximation. Int. Math. Res. Not. IMRN 2010(1) (2010), 18–52.Google Scholar
Huang, J.. Rational points near planar curves and Diophantine approximation. Adv. Math. 274 (2015), 490–515.Google Scholar
Huang, J.. The density of rational points near hypersurfaces. Duke Math. J. 169(11) (2020), 2045–2077.CrossRefGoogle Scholar
Huang, J.. Diophantine approximation on the parabola with non-monotonic approximation functions. Math. Proc. Camb. Phil. Soc. 168(3) (2020), 535–542.CrossRefGoogle Scholar
Huang, J. and Liu, J.. Simultaneous approximation on affine subspaces. Internat. Math. Res. Not. 19 (2018), 14905–14921.Google Scholar
Huxley, M. N.. Area, lattice points, and exponential sums, London Math. Soc. Monogr. N.S. vol. 13 (The Clarendon Press, Oxford University Press, New York, 1996). Oxford Science Publications.Google Scholar
Jarnik, V.. Diophantische approximationen und hausdorffsches mass. Mat. Sbornik 36 (1929), 371–382.Google Scholar
Jarník, V.. Sur les approximations diophantiques des nombres p-adiques. Rev. Ci. (Lima) 47 (1945), 489–505.Google Scholar
Khintchine, A.. Zur metrischen Theorie der diophantischen Approximationen. Math. Z. 24(1) (1926), 706–714.CrossRefGoogle Scholar
Kleinbock, D. and Margulis, G.. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2), 148(1) (1998), 339–360.CrossRefGoogle Scholar
Kleinbock, D. and Tomanov, G.. Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comment. Math. Helv. 82(3) (2007), 519–581.CrossRefGoogle Scholar
Kochen, S. and Stone, C.. A note on the Borel-Cantelli lemma. Illinois J. Math. 8 (1964), 248–251.CrossRefGoogle Scholar
Lutz, E.. Sur les approximations diophantiennes linéaires P-adiques. Actualités Sci. Ind., no. 1224 (Hermann and Cie, Paris, 1955).Google Scholar
Mahler, K.. Introduction to p-adic Numbers and their Functions. Cambridge Tracts in Math. no. 64 (Cambridge University Press, London-New York, 1973).Google Scholar
Mohammadi, A. and Golsefidy, A. S.. Simultaneous Diophantine approximation in non-degenerate p-adic manifolds. Israel J. Math. 188 (2012), 231–258.CrossRefGoogle Scholar
Ramrez, F.. Khintchine types of translated coordinate hyperplanes. Acta Arith. 170(3) (2015), 243–273.CrossRefGoogle Scholar
Rynne, B.. Hausdorff dimension and generalised simultaneous Diophantine approximation. Bull. London Math. Soc. 30(4) (1998), 365–376.CrossRefGoogle Scholar
Schikhof, W.. Ultrametric Calculus Camb. Stud. Adv. Math. vol. 4 (Cambridge University Press, Cambridge, 2006). An introduction to p-adic analysis, Reprint of the 1984 original [MR0791759].CrossRefGoogle Scholar
Schleischitz, J.. On the spectrum of Diophantine approximation constants. Mathematika 62(1) (2016), 79–100.CrossRefGoogle Scholar
Schleischitz, J.. Diophantine approximation on polynomial curves. Math. Proc. Camb. Phil. Soc. 163(3) (2017), 533–546.CrossRefGoogle Scholar
Schneider, P.. p-adic Lie Groups Grundlehren Math. Wiss. vol. 344 (Springer, Heidelberg, 2011).CrossRefGoogle Scholar
Simmons, D.. Some manifolds of Khinchin type for convergence. J. Théor. Nombres Bordeaux 30(1) (2018), 175–193.CrossRefGoogle Scholar
Sprindžuk, V.. Mahler’s problem in metric number theory . Translated from the Russian by B. Volkmann. Trans. Math. Monogr. vol. 25 (American Mathematical Society, Providence, R.I., 1969).Google Scholar
Sprindžuk, V.. Metric Theory of Diophantine Approximations (V. H. Winston and Sons, Washington, D.C.; A Halsted Press Book, John Wiley and Sons, New York-Toronto, Ont.-London, 1979). Translated from the Russian and edited by Richard A. Silverman, With a foreword by Donald J. Newman, Scripta Series in Mathematics.Google Scholar
Sprindžuk, V.. Achievements and problems of the theory of Diophantine approximations. Uspekhi Mat. Nauk 35(4(214)) (1980), 3–68, 248.CrossRefGoogle Scholar
Tricot, C.. Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc. 91(1) (1982), 57–74.CrossRefGoogle Scholar
Vaughan, R. C. and Velani, S.. Diophantine approximation on planar curves: the convergence theory. Invent. Math. 166(1) (2006), 103–124.CrossRefGoogle Scholar
Wang, B. and Wu, J.. Mass transference principle from rectangles to rectangles in Diophantine approximation. Math. Ann. 381(1-2) (2021), 243–317.CrossRefGoogle Scholar
Wang, B., Wu, J. and Xu, J.. Mass transference principle for limsup sets generated by rectangles. Math. Proc. Camb. Phil. Soc. 158(3) (2015), 419–437.CrossRefGoogle Scholar