Remarks on the Pair Correlation Statistic of Kronecker Sequences and Lattice Point Counting

TL;DR

This paper reformulates the pair correlation analysis of Kronecker sequences as a lattice point counting problem, applying recent techniques to establish $eta$-pair correlations for a class of these sequences.

Contribution

It introduces a lattice point counting approach to analyze pair correlations of Kronecker sequences, extending methods used for the three gap property.

Findings

Kronecker sequence pair correlation can be reformulated as lattice point counting

Recent lattice techniques show certain Kronecker sequences have $eta$-pair correlations for all $0<eta<1$

Provides a new perspective connecting sequence statistics with lattice geometry

Abstract

In this short note, we reformulate the task of calculating the pair correlation statistics of a Kronecker sequence as a lattice point counting problem. This can be done analogously to the lattice based approach which was used to (re-)prove the famous three gap property for Kronecker sequences. We show that recently developed lattice point counting techniques can then be applied to derive that a certain class of Kronecker sequences have $\beta$-pair correlations for all $0 < \beta < 1$.