The distribution of alpha p modulo one

: Matomaki K

Publisher: CAMBRIDGE UNIV PRESS

: 2009

: Mathematical Proceedings of the Cambridge Philosophical Society

: MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY

: MATH PROC CAMBRIDGE

: 147

: 267

: 283

: 17

: 0305-0041

DOI: https://doi.org/10.1017/S030500410900245X

We prove that, for any irrational number alpha, there are infinitely many primes p such that ||alpha p|| < p^(-1/3+epsilon). 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.

alphapmod1.pdf