A note on palindromic length of Sturmian sequences

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 $P{L}_u$, 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 ${\mathrm{limsup}}_{n\to \infty }P{L}_u(n)=+\infty$ for any Sturmian word $u$. We show that there is a constant $K>0$ such that $P{L}_u(n)\le K\ln n$ 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 $P{L}_u(n)=O(f(n))$.