Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average

Abstract

We show that, for sufficiently large integersm and X, for almost all a = 1,...,m the ratios a/x and the products ax, where |x| 6 X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums