Distinction of measures of Haar cylinders in the Dirichlet theorem for the field of p-adic numbers

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.