An inequality for the number of periods in a word

TL;DR

This paper establishes an inequality relating the number of periods in a word to its length and critical exponent, characterizes periods in Sturmian words, and explores measures of periodicity and special cases like overlap-free words.

Contribution

It introduces a new inequality for periods in words, characterizes periods in Sturmian words via Ostrowski representation, and proposes measures of periodicity for infinite words.

Findings

Inequality for the number of periods in words

Characterization of periods in Sturmian words

Tightness of the inequality for infinitely many words

Abstract

We prove an inequality for the number of periods in a word x in terms of the length of x and its initial critical exponent. Next, we characterize all periods of the length-n prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of n, and use this result to show that our inequality is tight for infinitely many words x. We propose two related measures of periodicity for infinite words. Finally, we also consider special cases where x is overlap-free or squarefree.