Avoiding Two Consecutive Blocks of Same Size and Same Sum over $\mathbb{Z}^2$
Michaël Rao, Matthieu Rosenfeld · SIAM Journal on Discrete Mathematics · 2018
A long standing question asks whether $\mathbb{Z}$ is uniformly 2-repetitive, that is, whether or not there is an infinite sequence over a finite subset of $\mathbb{Z}$ avoiding two consecutive blocks of the same size and same sum [J. Justin, J. Combin. Theory Ser. A, 12 (1972), pp. 357--367], [G. Pirillo and S. Varricchio, Semigroup Forum, 49 (1994), pp. 125--129]. Cassaigne et al. [ J. ACM, 61 (2014), 10] 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 a weak version of it.