In this paper, together with  and , we shall be concerned with estimating the number of solutions of the inequality
for almost all α (in the sense of Lebesgue measure on Iℝ), where , and both m and n are restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in  of an earlier result of Khintchine ) 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 sum
diverges, or converges, respectively.