One-dimensional quasi-uniform Kronecker sequences

Takashi Goda

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arXiv · 2404.16352

One-dimensional quasi-uniform Kronecker sequences

Takashi Goda

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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).

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Takashi Goda. 2024-04-25. One-dimensional quasi-uniform Kronecker sequences. https://arxiv.org/abs/2404.16352

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