On the pair correlations of powers of real numbers
Christoph Aistleitner, Simon Baker

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 the sequence 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 the pair correlations of the fractional parts of are asymptotically Poissonian. The proof is based on a martingale approximation method.
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