A note on palindromic length of Sturmian sequences

TL;DR

This paper investigates the growth of the palindromic length of factors in Sturmian sequences, establishing an upper logarithmic bound and demonstrating the existence of Sturmian words with arbitrarily slow-growing palindromic length functions.

Contribution

It proves a universal logarithmic upper bound for palindromic length in Sturmian words and constructs examples with controlled growth rates.

Findings

Palindromic length in Sturmian words is bounded above by a constant times log n.

There exist Sturmian words with palindromic length growth rates matching any unbounded non-decreasing function.

The limsup of palindromic length for Sturmian words diverges to infinity.

Abstract

Frid, Puzynina and Zamboni (2013) defined the palindromic length of a finite word $w$ as the minimal number of palindromes whose concatenation is equal to $w$. For an infinite word $u$ we study $PL_{u}$, that is, the function that assigns to each positive integer $n$, the maximal palindromic length of factors of length $n$ in $u$. Recently, Frid (2018) proved that $\limsup_{n\to\infty} PL_{u}(n)=+\infty$ for any Sturmian word $u$. We show that there is a constant $K>0$ such that $PL_{u}(n)\leq K\ln n$ for every Sturmian word $u$, and that for each non-decreasing function $f$ with property $\lim_{n\to\infty}f(n)=+\infty$ there is a Sturmian word $u$ such that $PL_{u}(n)=O(f(n))$.