Avoiding two consecutive blocks of same size and same sum over ${\mathbb{Z}}^2$

Abstract

A long standing question asks whether $\mathbb{Z}$ is uniformly 2-repetitive [Justin 1972, Pirillo and Varricchio, 1994], that is, whether there is an infinite sequence over a finite subset of $\mathbb{Z}$ avoiding two consecutive blocks of same size and same sum or not. Cassaigne \emph{et al.} [2014] showed that $\mathbb{Z}$ is not uniformly 3-repetitive. We show that ${\mathbb{Z}}^2$ is not uniformly 2-repetitive. Moreover, this problem is related to a question from Mäkelä in combinatorics on words and we answer to a weak version of it.