About density of closed factors in infinite words

Abstract: 

A word is a sequence of symbols from an finite alphabet. A finite word is called closed if it has length ≤1 or it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences. For example, the words abab, ababa are closed. The number of distinct closed factors of w is denoted by Cl(w).

An infinite word u is called closed-rich if there exists a positive constant C such that any factor of u of length n ≥1 contains at least C n^2 distinct closed factors. Here C is called the closed-rich constant of u. In our studies, we consider that if u is a closed-rich infinite word, then its closed-rich constant is C = inf{ min{Cl(w): w∈Σ^n ∩ Fac(u)} / n^2 : n ≥1}. Basically, C measures the density of closed factors in any factor of u.

Parshina and Puzynina (Theo. com. Sci., 2024) showed that C ≤1/6 for any infinite closed-rich word and asked a question about the value of Csup = sup{C : C is the closed-rich constant of infinite closed-rich word u}.

In this talk, I will present an improved bound of C for any infinite closed-rich word, along with a bound for Csup using the Fibonacci sequence.

The talk is based on joint work with Prof. Svetlana Puzynina.