Limiting behavior of Rudin-Shapiro sequence autocorrelations

Abstract: The Rudin-Shapiro polynomials $p_{m}$ were first studied by Rudin, Shapiro, and Golay independently nearly 80 years ago and are defined recursively by $p_{0}(x) = q_{0}(x) = 1$ and $$ p_{m}(x) = p_{m-1}(x) + x^{2^{m-1}} q_{m-1}(x), $$ $$ q_{m}(x) = p_{m-1}(x) - x^{2^{m-1}} q_{m-1}(x). $$ This class of polynomials benefits from rich structure and is of special interest as a subset of Littlewood polynomials (i.e., polynomials with coefficients in $\{ -1, 1 \}$), in part due to their having small $L^{4}$ norm, which is desirable and almost always unsatisfied by Littlewood polynomials in general. In application, uses are found for the Rudin-Shapiro polynomials in varied contexts such as radio and spectroscopy.

If we let $p_{m}(x) = \sum_{j=0}^{2^{m}-1} a_{j}x^{j}$, the sequence $(a_{0}, a_{1}, \dots, a_{2^{m}-1})$ is called the $m$-th Rudin-Shapiro sequence. We denote by $C_{m}(k)$ the aperiodic autocorrelation at shift $k$ of the $m$-th Rudin-Shapiro sequence: $$ C_{m}(k) = \sum_{j=0}^{2^{m}-1} a_{j}a_{j+k}, $$ where it is understood that $a_{j} = 0$ for $j \notin [0, 2^{m}-1]$. These autocorrelations have been studied extensively. It is often difficult to determine or approximate $C_{m}(k)$ for any given $m$ and $k$, but using the structure of $p_{m}$, bounds on partial moments of the $C_{m}(k)$ can be deduced. We give the precise orders of $\sum_{0 < k \leq x} (C_{m}(k))^{2} $ and $\max_{0 < k \leq x} |C_{m}(k)|$, and asymptotic bounds for $\sum_{0 < k \leq x} |C_{m}(k)|$. Furthermore, we construct an analogue of $|C_{m}(k)|$ on $[0,1]$ and show that its maximum value occurs uniquely at $x = \frac{2}{3}$, supporting our conjecture that the maximum value of $|C_{m}(k)|$ occurs uniquely at some $k_{m}^{\ast}$ with $\lim_{m \to \infty} \frac{k_{m}^{\ast}}{2^{m}} = \frac{2}{3}$. This is joint work with Stephen Choi.

algebraic geometrynumber theory

Audience: researchers in the discipline

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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