On combinatorial properties of the Arshon sequence

We consider combinatorial and algebraic properties of the language of factors of the in+nite sequence on the three-letter alphabet built by S.E. Arshon in 1930s. This sequence never contains two successive equal words, i. e., avoids the number 2. The notion of avoidability is extended from integers to rational numbers. It is shown that the avoidability bound for the considered language is 4 . This language is de+ned by two alternating morphisms; our method allows to study it like a formal language de+ned by one morphism. We also give a complete description of the syntactic congruence of the considered language. ? 2001 Elsevier Science B.V. All rights reserved.