On the pair correlations of powers of real numbers

摘要

A classical theorem of Koksma states that for Lebesgue almost every x > 1 the sequence (x(n))n=1(infinity) 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(infinity) are asymptotically Poissonian. The proof is based on a martingale approximation method.