A Metrical Theorem in Diophantine Approximation

Abstract

In this paper we prove a sharpening and generalization of the following Theorem of Khintchine (4):Let ψ1(q), …, ψnq) be n non-negative junctions of the positive integer q and assumeis monotonically decreasing. Then the set of inequalities1has an infinity of integer solutions q > 0 and p1, … , pn for almost all or no sets of numbers θ1, … , θ2, according as Σψ(q) diverges or converges.