A Metrical Theorem in Diophantine Approximation

Abstract

In this paper we prove a sharpening and generalization of the following Theorem of Khintchine (4): Let ψ 1 (q) , …, ψ n q) be n non-negative junctions of the positive integer q and assume is monotonically decreasing. Then the set of inequalities 1 has an infinity of integer solutions q > 0 and p 1 , … , p n for almost all or no sets of numbers θ 1 , … , θ 2 , according as Σψ(q) diverges or converges.