Remarks on the pair correlation statistic of Kronecker sequences and lattice point counting

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 $$\beta $$ β -pair correlations for all $$0< \beta < 1$$ 0 < β < 1 .