SIMULTANEOUS AVOIDANCE OF LARGE SQUARES AND FRACTIONAL POWERS IN INFINITE BINARY WORDS

Abstract

In 1976, Dekking showed that there exists an infinite binary word that contains neither squares yy with |y|≥4 nor cubes xxx. We show that 'cube' can be replaced by any fractional power > 5/2. We also consider the analogous problem where '4' is replaced by any integer. This results in an interesting and subtle hierarchy.