Metric Diophantine approximation with two restricted variables I. Two square-free integers, or integers in arithmetic progressions

Abstract

In this paper, together with [7] and [8], we shall be concerned with estimating the number of solutions of the inequalityfor almost all α (in the sense of Lebesgue measure on Iℝ), where, and bothmandnare restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in [2] of an earlier result of Khintchine [10]) and its quantitative developments (for example, see [11, 12,6]):Let ψ(n) be a non-increasing positive function of a positive integer variable n. Then the inequality (1·1) has infinitely many, or only finitely many, solutions in integers to, n(n > 0) for almost all real α, according to whether the sumdiverges, or converges, respectively.