Frequencies of letters in infinite k-balanced sequences

The frequency of letters in a symbolic sequence u over a finite alphabet is one of the basic characteristics of u. The notion of k-balancedness captures the property that the number of any letter occurring in two arbitrary factors of u of equal length differs at most by k. For a fixed integer k and alphabet size d ∈ 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 provide 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.