One-dimensional quasi-uniform Kronecker sequences

TL;DR

This paper proves that one-dimensional Kronecker sequences are quasi-uniform precisely when the parameter is badly approximable, using elementary methods and the three-gap theorem.

Contribution

It establishes a necessary and sufficient condition for quasi-uniformity of Kronecker sequences in one dimension, linking it to badly approximable numbers.

Findings

Kronecker sequence is quasi-uniform iff α is badly approximable

Elementary proof based on the three-gap theorem

Clarifies the structure of distribution for these sequences

Abstract

In this short note, we prove that the one-dimensional Kronecker sequence $i\alpha \bmod 1, i=0,1,2,\ldots,$ is quasi-uniform if and only if $\alpha$ is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (1965).