On the pair correlations of powers of real numbers

Abstract

A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n{)}_{n=1}^{\infty }$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n{)}_{n=1}^{\infty }$ are asymptotically Poissonian. The proof is based on a martingale approximation method.

Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics