On the pair correlations of powers of real numbers

TL;DR

This paper extends Koksma's theorem to show that for almost every real number greater than one, the pair correlations of the fractional parts of its powers are asymptotically Poissonian, indicating a form of randomness.

Contribution

It introduces a novel extension of Koksma's theorem to pair correlations, demonstrating Poissonian behavior for almost all such sequences.

Findings

Pair correlations are asymptotically Poissonian for almost all x>1.

The proof employs a martingale approximation method.

Extends classical uniform distribution results to pair correlation setting.

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.