On a Diophantine Inequality with Reciprocals

Abstract

A sharpened lower bound is obtained for the number of solutions to an inequality of the form α ≤ {(an̅ + bn)/q} < β, 1 ≤ n ≤ N, where q is a sufficiently large prime number, a and b are integers with (ab, q) = 1, nn̅ ≡ 1 (mod q), and 0 ≤ α < β ≤ 1. The length N of the range of the variable n is of order q^ε, where ε > 0 is an arbitrarily small fixed number.