A Note on the Abelian Complexity of the Rudin-Shapiro Sequence

Abstract

Let {r(n)}n≥0 be the Rudin-Shapiro sequence, and let ρ(n):=max{∑j=ii+n−1r(j)∣i≥0}+1 be the abelian complexity function of the Rudin-Shapiro sequence. In this note, we show that the function ρ(n) has many similarities with the classical summatory function Sr(n):=∑i=0nr(i). In particular, we prove that for every positive integer n, 3≤ρ(n)n≤3. Moreover, the point set {ρ(n)n:n≥1} is dense in [3,3].

Keywords

• Rudin-Shapiro sequence
• abelian complexity
• growth order
• dense property