Frequencies of letters in infinite $k$-balanced sequences

Abstract

Frequency of letters in a symbolic sequence $\mathbf{u}$ over a finite alphabet is one of the basic characteristics of $\mathbf{u}$. The notion of $k$-balancedness captures the property that the number of any letter occurring in two arbitrary factors of $\mathbf{u}$ of equal length differs at most by $k$. For a fixed integer $k$ and alphabet size $d\in \mathbb{N}$, we discuss possible frequencies of letters in $k$-balanced $d$-ary sequences. For the size $d$ of the alphabet, we introduce the notion of balancedness threshold $BT(d)$ and give an upper bound on it, where $BT(d)$ is the minimum $k$ such that there exists a $k$-balanced sequence over a $d$-letter alphabet for all possible letter frequencies.