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

TL;DR

This paper proves that the two-dimensional integer lattice $ olinebreak bZ^2$ cannot be uniformly 2-repetitive, addressing a longstanding question in combinatorics on words and extending previous results from the one-dimensional case.

Contribution

It establishes that $bZ^2$ is not uniformly 2-repetitive, providing a significant extension of known results from $bZ$ and connecting to open problems in combinatorics on words.

Findings

$bZ^2$ is not uniformly 2-repetitive

Addresses a long-standing open problem in combinatorics on words

Provides a partial answer to a question posed by M"akel"a

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 M\"akel\"a in combinatorics on words and we answer to a weak version of it.