One-dimensional quasi-uniform Kronecker sequences

Abstract

In this short note, we prove that the one-dimensional Kronecker sequence $i\alpha \mathrm{mod}1,i=0,1,2,\dots ,$ 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).

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