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

Xiaotao Lü and Pengju Han
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Xiaotao Lü: College of Science, Huazhong Agricultural University, Wuhan 430070, China
Pengju Han: College of Science, Huazhong Agricultural University, Wuhan 430070, China

Mathematics, 2022, vol. 10, issue 2, 1-10

Abstract: Let { r ( n ) } n ≥ 0 be the Rudin-Shapiro sequence, and let ρ ( n ) : = max { ∑ j = i i + n − 1 r ( 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 S r ( n ) : = ∑ i = 0 n r ( 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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