UNIMPOVABILITY OF DIRICHLET’S THEOREM FOR POLYNOMIALS

Abstract

The article relates to the classes of S-numbers in Mahler’s classification [1]. There are a number of the wellknown results on finding lower bounds in the theory of Diophantine equations. Some of them include a lower bound for approximation of rational numbers; a lower bound for approximation of algebraic numbers obtained by Liouville; a later improvement of the result of Liouville known as the Thue – Siegel – Roth theorem [2]. However, the bounds described above are considered ineffective in the sense that their proof does not give a way how to calculate them. In this case, these results and their proofs cannot be used to estimate the magnitude of the solutions of the corresponding Diophantine equations, but can be used to estimate the number of solutions of these equations. In the article, using the methods of metric theory of Diophantine approximations, we have considered individual and global lower bounds for polynomials [3]. A new global bound has been obtained for the unimprovability of Dirichlet’s theorem using the metric approach for finding a lower bound in a given interval for polynomials of a degree of no more than n and an additional condition for the modulus of the derivative of this polynomial.