Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences

TL;DR

This paper improves classical theorems related to fractional parts of multiples of badly approximable irrationals, leading to better understanding of sequence diversity in Sturmian sequences with such slopes.

Contribution

It provides an optimal version of the three-gap theorem and a Kronecker approximation result specifically for badly approximable numbers, with applications to Sturmian sequences.

Findings

Enhanced three-gap theorem for badly approximable numbers

Improved inhomogeneous Kronecker approximation for these numbers

Better measure of sequence diversity in Sturmian sequences

Abstract

We give an optimal version of the classical ``three-gap theorem'' on the fractional parts of $n \theta$, in the case where $\theta$ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker's inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.