Balanced words in higher dimensions

Abstract

For $d\ge 1$, a word $w\in \{0,1{\}}^{{\mathbb{Z}}^d}$ is called balanced if there exists $M>0$ such that for any two rectangles $R,R^{{}^{\prime }}\subset {\mathbb{Z}}^d$ that are translates of each other, the number of occurrences of the symbol $1$ in $R$ and $R^{{}^{\prime }}$ differ by at most $M$. It is known that for every balanced word $w$, the asymptotic frequency of the symbol $1$ ( called the density of $w$ ) exists. In this paper we show that there exist two dimensional balanced words with irrational densities, answering a question raised by Berthé and Tijdeman.

References in corpus (1)

• Palindromic Richness