On a theorem of V. Bernik in the metrical theory of Diophantine approximation

TL;DR

This paper extends Bernik's convergence Khintchine theorem in metrical Diophantine approximation by removing the previously required monotonicity condition, thereby broadening its applicability.

Contribution

It removes the monotonicity assumption in Bernik's theorem, providing a more general result in Diophantine approximation by polynomials.

Findings

Monotonicity assumption is eliminated from Bernik's theorem.

The generalized theorem applies to arbitrary error functions.

Results improve the scope of metrical Diophantine approximation theories.

Abstract

This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a convergence Khintchine type theorem for Diophantine approximation by polynomials, that is it allows arbitrary monotonic error of approximation. In the present paper the monotonicity assumption is completely removed.