The distribution of αp modulo one

Abstract

AbstractWe prove that, for any irrational number α, there are infinitely many primes p such that ∥αp∥ < p−1/3+ε. Here ∥y∥ denotes the distance from y to the nearest integer. The proof uses Harman's sieve method with arithmetical information coming from bounds for averages of Kloosterman sums.