Limiting behavior of Rudin–Shapiro sequence autocorrelations

Let ( r 0 , r 1 , ⋯ , r 2 m − 1 ) (r_0, r_1, ⋯ , r2^m-1) be the m m th Rudin–Shapiro sequence defined by \[ r i = ( − 1 ) # of pairs of consecutive ones in the binary expansion of i rᵢ = (-1)^{# of pairs of consecutive ones in the binary expansion of i} \] and C m ( k ) C_m(k) be its aperiodic autocorrelation at shift k k . In this paper, we study the ℓ 1 ^1 , ℓ 2 ² , and ℓ ∞ ∞ moments of C m ( k ) Cₘ(k) and prove that ∑ 0 > k ≤ x ( C m ( k ) ) 2 ∑ 0 > k ≤ x (Cₘ(k))^2 and max 0 > k ≤ x | C m ( k ) | max 0 > k ≤ x |Cₘ(k)| are of the same order as x 2 x² and x log 2 ⁡ ( λ ) x^{log ₂(λ )} , respectively, where λ λ is the real root of x 3 + x 2 − 2 x − 4 x³ + x^2 - 2x - 4 . We also give asymptotic upper and lower bounds for ∑ 0 > k ≤ x | C m ( k ) | ∑ 0 > k ≤ x |Cₘ(k)| . These findings generalize results of Littlewood, Choi, Katz and van der Linden, and Allouche et al. Analogous results for periodic autocorrelation are also obtained. Furthermore, we construct and study a continuous analogue of C m ( k ) Cₘ(k) with domain [ 0 , 1 ] [0,1] and show that the maximum value of this function occurs uniquely at x = 2 / 3 x = 2/3 , which supports our conjecture that the maximum value of | C m ( k ) | |Cₘ(k)| occurs uniquely at some k m ∗ kₘ with lim m → ∞ k m ∗ 2 m = 2 3 lim m → ∞ {kₘ}{2ᵐ} = 2/3 .