Finite and infinite closed-rich words

TL;DR

This paper investigates words rich in closed factors, establishing asymptotic bounds for finite words and characterizing infinite closed-rich words, including Sturmian and linearly recurrent words, with several necessary and sufficient conditions.

Contribution

It introduces the concept of infinite closed-rich words, provides asymptotic analysis for finite words, and characterizes classes of infinite words that are closed-rich, such as Sturmian and linearly recurrent words.

Findings

Maximal number of closed factors in finite words grows as n^2/6

Existence of infinite closed-rich words with quadratic number of closed factors in each factor

All linearly recurrent words are closed-rich

Abstract

A word is called closed if it has a prefix which is also its suffix and there is no internal occurrences of this prefix in the word. In this paper we study words that are rich in closed factors, i.e., which contain the maximal possible number of distinct closed factors. As the main result, we show that for finite words the asymptotics of the maximal number of distinct closed factors in a word of length $n$ is $\frac{n^2}{6}$. For infinite words, we show there exist words such that each their factor of length $n$ contains a quadratic number of distinct closed factors, with uniformly bounded constant; we call such words infinite closed-rich. We provide several necessary and some sufficient conditions for a word to be infinite closed rich. For example, we show that all linearly recurrent words are closed-rich. We provide a characterization of rich words among Sturmian words. Certain examples we provide involve non-constructive methods.