One-dimensional quasi-uniform Kronecker 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 over the unit interval [0, 1] 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 (Proc Camb Philos Soc, 61:665–670, 1965).