On the maximal number of highly periodic runs in a string

TL;DR

This paper investigates highly periodic runs in strings, establishing an upper bound of 0.5n and a lower bound of 0.406n for their maximum count in strings of length n, advancing understanding of string periodicity.

Contribution

It introduces bounds on the number of highly periodic runs in strings, a new focus compared to previous studies on general runs.

Findings

Upper bound of 0.5n on highly periodic runs

Constructed sequence with at least 0.406n highly periodic runs

Advances understanding of string periodicity constraints

Abstract

A run is a maximal occurrence of a repetition $v$ with a period $p$ such that $2p \le |v|$. The maximal number of runs in a string of length $n$ was studied by several authors and it is known to be between $0.944 n$ and $1.029 n$. We investigate highly periodic runs, in which the shortest period $p$ satisfies $3p \le |v|$. We show the upper bound $0.5n$ on the maximal number of such runs in a string of length $n$ and construct a sequence of words for which we obtain the lower bound $0.406 n$.