Balanced rectangles over Sturmian words and minimal discrepancy intervals

TL;DR

This paper characterizes the balance properties of matrices formed from Sturmian words with slope alpha using Ostrowski representations, extending previous work to all irrational slopes through distribution analysis.

Contribution

It generalizes prior results by providing a complete characterization for all irrational slopes using Ostrowski representations and distribution of n alpha mod 1.

Findings

Characterization of balance properties in terms of Ostrowski representations

Extension of results to all irrational slopes, not just quadratic irrationals

Use of distribution of n alpha mod 1 in analysis

Abstract

We consider $m\times n$ rectangular matrices formed from Sturmian words with slope $\alpha$, and we fully characterise their balance properties in terms of the Ostrowski representations of $m$ and $n$ with respect to $\alpha$. This generalises recent results by Anselmo et al., as well as those by Shallit and the author, where only quadratic irrational slopes were considered. In contrast to the two mentioned papers, the approach in this paper is based on the distribution of $n\alpha \bmod 1$.