FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
Distinction of measures of Haar cylinders in the Dirichlet theorem for the field of p-adic numbers |
Bernik V. I., Kudin A. S., Titova A. V. |
2023, issue 1, P. 3-11 DOI: https://doi.org/10.47910/FEMJ202301 |
Abstract |
| The Dirichlet box principle gives surprisingly accurate results in problems of approximation of real numbers by rational numbers, transcendental numbers by real algebraic numbers. Every polynomial taking small values at a given point $x$ also takes small values in its neighborhood. A problem of studying such neighborhoods and obtaining possible Lebesgue measure values arises frequently. In this paper we solve the problem in the p-adic case using recent results of the metric theory of Diophantine approximations. |
Keywords: Diophantine approximations, Haar measure, p-adic numbers, Dirichlet theorem. |
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References |
| [1] V. G. Sprindzhuk, Problema Malera v metricheskoj teorii chisel, Moskva, 1967. [2] V. G. Sprindzhuk, Metricheskaja teorija diofantovyh priblizhenij, Moskva, 1977. [3] B. Volkmann, “The real cubic case of Mahler’s conjecture”, Mathematika, 1961. [4] V. Bernik, F. Gotze, Distribution of real algebraic numbers of arbitrary degree in short intervals. V. 79, 2015. [5] K. Mahler, “Uber das Mass der Menge aller S-Zahlen”, Mathematische Annalen, 1932. [6] V. I. Bernik, M. M. Dodson, “Metric Diophantine approximation on manifolds”, Cambridge Tracts in Math, 1999. [7] A. Khintchine, “Einige satzeuber kettenbruche, mit anwendungen auf die theorie der Diophantischen approximationen”, Mathematische Annalen, 92 (1924). [8] V. I. Bernik, “O tochnom porjadke priblizhenija nulja znachenijami celochislennyh mnogochlenov”, Acta Arithmetica, 53 (1989). [9] V. Beresnevich, “On approximation of real numbers by real algebraic numbers”, Acta Arithmetica, 1999. [10] V. Beresnevich, “A Groshev type theorem for convergence on manifolds”, Acta Mathematica Hungarica, 2002. [11] V. Bernik, D. Kleinbock, G. Margulis, “Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions”, International Mathematics Research Notices, 2001. [12] Y. Bugeaud, “On the approximation to algebraic numbers by algebraic numbers”, Cambridge Tracts in Math, 2004. [13] V. V. Beresnevich, V. I. Bernik, E. I. Kovalevskaya, “On approximation of p-adic numbers by p-adic algebraic numbers”, Journal of Number Theory, 111 (2005). [14] V. I. Bernik, I. L. Morockaja, “Diofantovy priblizhenija v Qp i razmernost' Hausdorfa”, Vesci AN BSSR. Ser. fiz.-mat. navuk, 1986. [15] V. I. Bernik, I. A. Korljukova, A. S. Kudin, A. V. Titova, “Celochislennye mnogochleny i teorema Minkovskogo o linejnyh formah”, Vestnik Grodnenskogo gosudarstvennogo universiteta imeni Janki Kupaly, 2022. |