A note on palindromic length of Sturmian sequences

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 $\limsup_{n\to\infty} PL_{u}(n)=+\infty$foranySturmianword$u$\mathrm{.}Weshowthatthereisaconstant$K>0$suchthat$PL_{u}(n)\leq K\ln n$foreverySturmianword$u$, and that for each non-decreasing function $f$ with property ${\lim }_{n\to \mathrm{\infty }}f(n)=+\mathrm{\infty }$ there is a Sturmian word $u$ such that $P{L}_u(n)=O(f(n))$.