Geodesic
Distinction of measures of Haar cylinders in the Dirichlet theorem for the field of p-adic numbers
Dalʹnevostočnyj matematičeskij žurnal, Tome 23 (2023) no. 1, pp. 3-11 
V. I. Bernik; A. S. Kudin; A. V. Titova. Distinction of measures of Haar cylinders in the Dirichlet theorem for the field of p-adic numbers. Dalʹnevostočnyj matematičeskij žurnal, Tome 23 (2023) no. 1, pp. 3-11. http://geodesic.mathdoc.fr/item/DVMG_2023_23_1_a0/
@article{DVMG_2023_23_1_a0,
     author = {V. I. Bernik and A. S. Kudin and A. V. Titova},
     title = {Distinction of measures of {Haar} cylinders in the {Dirichlet} theorem for the field of p-adic numbers},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {3--11},
     year = {2023},
     volume = {23},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2023_23_1_a0/}
}
TY  - JOUR
AU  - V. I. Bernik
AU  - A. S. Kudin
AU  - A. V. Titova
TI  - Distinction of measures of Haar cylinders in the Dirichlet theorem for the field of p-adic numbers
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2023
SP  - 3
EP  - 11
VL  - 23
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DVMG_2023_23_1_a0/
LA  - ru
ID  - DVMG_2023_23_1_a0
ER  - 
%0 Journal Article
%A V. I. Bernik
%A A. S. Kudin
%A A. V. Titova
%T Distinction of measures of Haar cylinders in the Dirichlet theorem for the field of p-adic numbers
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2023
%P 3-11
%V 23
%N 1
%U http://geodesic.mathdoc.fr/item/DVMG_2023_23_1_a0/
%G ru
%F DVMG_2023_23_1_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] V. G. Sprindzhuk, Problema Malera v metricheskoi teorii chisel, Moskva, 1967 | MR

[2] V. G. Sprindzhuk, Metricheskaya teoriya diofantovykh priblizhenii, Moskva, 1977

[3] B. Volkmann, “The real cubic case of Mahler’s conjecture”, Mathematika, 136:5 (1961) | MR

[4] V. Bernik, F. Götze, Distribution of real algebraic numbers of arbitrary degree in short intervals, v. 79, 2015 | MR

[5] K. Mahler, “Uber das Mass der Menge aller $S$-Zahlen”, Mathematische Annalen, 106 (1932) | DOI | MR | Zbl

[6] V. I. Bernik, M. M. Dodson, “Metric Diophantine approximation on manifolds”, Cambridge Tracts in Math., 1999, Cambridge | MR

[7] A. Khintchine, “Einige sätze über kettenbrüche, mit anwendungen auf die theorie der Diophantischen approximationen”, Mathematische Annalen, 92 (1924) | DOI | MR

[8] V. I. Bernik, “O tochnom poryadke priblizheniya nulya znacheniyami tselochislennykh mnogochlenov”, Acta Arithmetica, 53 (1989) | DOI | Zbl

[9] V. Beresnevich, “On approximation of real numbers by real algebraic numbers”, Acta Arithmetica, 90 (1999) | DOI | MR | Zbl

[10] V. Beresnevich, “A Groshev type theorem for convergence on manifolds”, Acta Mathematica Hungarica, 94 (2002) | DOI | MR | Zbl

[11] V. Bernik, D. Kleinbock, G. Margulis, “Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions”, International Mathematics Research Notices, 9 (2001) | MR | Zbl

[12] Y. Bugeaud, “On the approximation to algebraic numbers by algebraic numbers”, Cambridge Tracts in Math, 169 (2004), Cambridge | MR

[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) | DOI | MR | Zbl

[14] V. I. Bernik, I. L. Morotskaya, “Diofantovy priblizheniya v $Q_p$ i razmernost Khausdorfa”, Vestsi AN BSSR. Ser. fiz.-mat. navuk, 3 (1986) | MR | Zbl

[15] V. I. Bernik, I. A. Korlyukova, A. S. Kudin, A. V. Titova, “Tselochislennye mnogochleny i teorema Minkovskogo o lineinykh formakh”, Vestnik Grodnenskogo gosudarstvennogo universiteta imeni Yanki Kupaly, 2022