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 $\varvec{u}$ we study $\mathrm {pal}_{\varvec{u}}$, that is, the function that assigns to each positive integer n, the maximal palindromic length of factors of length n in $\varvec{u}$. Recently, Frid (2018) proved that $\limsup _{n\rightarrow \infty }\mathrm {pal}_{\varvec{u}}(n)=+\infty$ for any Sturmian word $\varvec{u}$. We show that there is a constant $K>0$ such that $\mathrm {pal}_{\varvec{u}}(n)\le K\ln n$ for every Sturmian word $\varvec{u}$, and that for each non-decreasing function f with property $\lim _{n\rightarrow \infty }f(n)=+\infty$ there is a Sturmian word $\varvec{u}$ such that $\mathrm {pal}_{\varvec{u}}(n)=\mathcal {O}(f(n))$.