Balanced rectangles over Sturmian words and minimal discrepancy intervals

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 $n\alpha 1$.