# Balanced words in higher dimensions

## TL;DR

This paper investigates higher-dimensional balanced words, demonstrating the existence of two-dimensional balanced words with irrational densities, thus answering a previously open question in the field.

## Contribution

It proves that balanced words in two dimensions can have irrational densities, extending the understanding of their properties beyond rational values.

## Key findings

- Existence of two-dimensional balanced words with irrational densities
- Answer to Berthé and Tijdeman's open question
- Extension of balanced word theory to 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.