Tight Upper Bounds on Distinct Maximal (Sub-)Repetitions in Highly Compressible Strings

Abstract

For [Formula: see text], maximal [Formula: see text]-repetitions ([Formula: see text]-subrepetitions) are fractional powers in strings with exponent of at least [Formula: see text] (and [Formula: see text], respectively) which are non-extendable with respect to their minimum period. In this paper, we show that in a string [Formula: see text] with string attractor [Formula: see text] there are at most [Formula: see text] distinct (unpositioned) extended maximal [Formula: see text]-repetitions. Also for any natural number [Formula: see text] the string contains at most [Formula: see text] distinct extended maximal [Formula: see text]-subrepetitions without [Formula: see text]th powers. We further prove that for fixed [Formula: see text] and [Formula: see text], both upper bounds are tight up to a constant factor.