k符号上无穷序列离散相关的界和广义Rudin-Shapiro序列

On the maximum order complexity of subsequences of the Thue-Morse and Rudin-Shapiro sequence along squares

Automatic sequences such as the Thue-Morse sequence and the Rudin-Shapiro sequence are highly predictable and thus not suitable in cryptography. In particular, they have small expansion complexity. However, they still have a large maximum order complexity. Certain subsequences of automatic sequences are not automatic anymore and may be attractive candidates for applications in cryptography. In this paper we show that subsequences along the squares of certain pattern sequences including the Thue-Morse sequence and the Rudin-Shapiro sequence have also large maximum order complexity but do not su…

Bounds on Rudin-Shapiro polynomials of arbitrary degree

Let $P_{<n}(z)$ be the Rudin-Shapiro polynomial of degree $n-1$. We show that $|P_{<n}(z)|\le \sqrt{6n-2}-1$ for all $n\ge0$ and $|z|=1$, confirming a longstanding conjecture. This bound is sharp in the case when $n=(2\cdot 4^k+1)/3$ and $z=1$. We also show that for $n\ge m\ge0$, $|P_{<n}(z)-P_{<m}(z)|\le \sqrt{10(n-m)}$, which is asymptotically sharp in the sense that for any $\varepsilon>0$ there exists $n>m\ge0$ and $z$ with $|z|=1$ and $|P_{<n}(z)-P_{<m}(z)|\ge\sqrt{(10-\varepsilon)(n-m)}$, contradicting a conjecture of Montgomery.

Integrals Involving Rudin-Shapiro Polynomials and Sketch of a Proof of Saffari's Conjecture

Continuing pioneering work of Christophe Doche and Laurent Habsieger from 2004, we develop computer algebra algorithms, implemented in Maple, for finding the (necessarily rational) generating function for any integral of products, and in particular, moments, of Rudin-Shapiro polynomials. We generate a lot of output, and confirm again a conjecture of Saffari for the asymptotics for small (and not so small) powers. We also confirm, for small powers, a related, more general, conjecture, of Hugh Montgomery. Finally, we outline a proof of Saffari's full conjecture, that we believe can be turned int…

Peak Sidelobe Level and Peak Crosscorrelation of Golay-Rudin-Shapiro Sequences

Sequences with low aperiodic autocorrelation and crosscorrelation are used in communications and remote sensing. Golay and Shapiro independently devised a recursive construction that produces families of complementary pairs of binary sequences. In the simplest case, the construction produces the Rudin-Shapiro sequences, and in general it produces what we call Golay-Rudin-Shapiro sequences. Calculations by Littlewood show that the Rudin-Shapiro sequences have low mean square autocorrelation. A sequence's peak sidelobe level is its largest magnitude of autocorrelation over all nonzero shifts. Hø…

广义Thue-Morse序列的形式逆与Rudin-Shapiro序列的变体

Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence

本文研究广义Thue-Morse序列及Rudin-Shapiro序列变体的形式逆，给出其递推关系与自动机，并分析连续相同字母串长度和字母频率，与原始序列比较。

The Rudin-Shapiro polynomials and The Fekete polynomials are not $L^α$-flat

We establish that the Rudin-Shapiro polynomials are not $L^α$-flat, for any $α\geq 0$. We further prove that the "truncated" Rudin-Shapiro sequence cannot generate a sequence of $L^α$-flat polynomials, for any $α\geq 0$. In the appendix, we present a simple proof of the fact that the Fekete polynomials and the modified or shifted Fekete polynomials are not $L^α$-flat, for any $α\geq 0$.

On the abelian complexity of the Rudin-Shapiro sequence

In this paper, we study the abelian complexity of the Rudin-Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function $ρ(n)$ which satisfies certain recurrence relations. As a consequence, the abelian complexity function is $2$-regular. Further, we prove that the box dimension of the graph of the asymptotic function $λ(x)$ is $3/2$ where $λ(x)=\lim_{k\to\infty}ρ(4^{k}x)/\sqrt{4^{k}x}$ and $ρ(x)=ρ(\lfloor x\rfloor)$ for any $x> 0$.

On the maximum order complexity of the Thue-Morse and Rudin-Shapiro sequence

Expansion complexity and maximum order complexity are both finer measures of pseudorandomness than the linear complexity which is the most prominent quality measure for cryptographic sequences. The expected value of the $N$th maximum order complexity is of order of magnitude $\log N$ whereas it is easy to find families of sequences with $N$th expansion complexity exponential in $\log N$. This might lead to the conjecture that the maximum order complexity is a finer measure than the expansion complexity. However, in this paper we provide two examples, the Thue-Morse sequence and the Rudin-Shapi…

On maximal autocorrelations of Rudin-Shapiro sequences

In this paper, we present an alternative proof showing that the maximal aperiodic autocorrelation of the $m$-th Rudin-Shapiro sequence is of the same order as $λ^{m}$, where $λ$ is the real root of $x^{3} + x^{2} - 2x - 4$. This result was originally proven by Allouche, Choi, Denise, Erdélyi, and Saffari (2019) and Choi (2020) using a translation of the problem into linear algebra. Our approach simplifies this linear algebraic translation and provides another method of dealing with the computations given by Choi. Additionally, we prove an analogous result for the maximal periodic autocorrelati…

Fibonacci、Thue-Morse 和 Rudin-Shapiro 词中的单色等差数列

Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words

本文研究 Fibonacci 词中固定公差最长单色等差数列的长度与位置，结合圆周旋转动力学与 Walnut 自动定理证明给出长度公式，并推广到 Thue-Morse 和 Rudin-Shapiro 词。

Discrete correlations of order 2 of generalised Rudin-Shapiro sequences: a combinatorial approach

We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the $n$th term of the sequence as being the total weight of the integer $n$ written in base $k$. Under an additional difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences, and we prove that they have the same correlations of order 2 as sequences of symbols chosen uniformly and independently at random. The speed of convergence is very fast and is independent of the prime factor decomposition …

Substitution-based sequences with absolutely continuous diffraction

Modifying Rudin's original construction of the Rudin-Shapiro sequence, we derive a new substitution-based sequence with purely absolutely continuous diffraction spectrum.