Balanced words in higher dimensions

Abstract

For $d\ge 1$, a word $w\in \{ 0,1\}^{\Z^d}$ is called balanced if there exists $M > 0$ such that for any two rectangles $R, R^{'}\subset\Z^d$ that are translates of each other, the number of occurrences of the symbol $1$ in $R$ and $R^{'}$ 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 Berth\'e and Tijdeman.