Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences
Abstract
We give an optimal version of the classical ``threegap 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 DOI:
 10.48550/arXiv.2006.15842
 arXiv:
 arXiv:2006.15842
 Bibcode:
 2020arXiv200615842B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics