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

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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. 
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     title = {Distinction of measures of {Haar} cylinders in the {Dirichlet} theorem for the field of p-adic numbers},
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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/