Avoiding two consecutive blocks of same size and same sum over ℤ²

2015/11/18 by Rao, Michaël, Rosenfeld, Matthieu · 1 citation
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL)

paper · doi:10.48550/arxiv.1511.05875

Abstract

A long standing question asks whether ℤ is uniformly 2-repetitive [Justin 1972, Pirillo and Varricchio, 1994], that is, whether there is an infinite sequence over a finite subset of ℤ avoiding two consecutive blocks of same size and same sum or not. Cassaigne et al. [2014] showed that ℤ is not uniformly 3-repetitive. We show that ℤ² 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.

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• Words with factor complexity 2n+1 and minimal critical exponent

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