Sequences modulo one: convergence of local statistics

TL;DR

This paper surveys recent advances in understanding the local statistical behavior of sequences modulo one, including angles in Euclidean lattices and the sequence of square roots modulo one, beyond classical equidistribution results.

Contribution

It compiles and discusses recent findings on the local statistics of specific sequences modulo one, highlighting developments beyond traditional equidistribution theory.

Findings

Analysis of angle sequences in Euclidean lattices

Results on the distribution of square roots modulo one

Insights into local statistical properties of these sequences

Abstract

We survey recent results beyond equidistribution of sequences modulo one. We focus on the sequence of angles in a Euclidean lattice in $\mathbb R^2$ and on the sequence $\sqrt n\bmod1 $.