Everywhere α-repetitive sequences and sturmian words

Abstract

Local constraints on an infinite sequence that imply global regularity are of general interest in combinatorics on words. We consider this topic by studying everywhere α-repetitive sequences, sequences in which every position has an occurrence of a repetition of order α > 1 of bounded length. The number of minimal such repetitions, called minimal α-powers, is then finite. A natural question regarding global regularity is to determine the least number of minimal α-powers such that an α-repetitive sequence is not necessarily ultimately periodic. We solve this question for 1 < α < 17/8. We also show that Sturmian words are among the optimal 2- and 2+-repetitive sequences. © Springer-Verlag Berlin Heidelberg 2007 Everywhere α-repetitive sequence, α-power, squareful sequence, overlapful sequence, Sturmian word.