Both the Thue-Morse and Rudin-Shapiro sequences are not suitable sequences for cryptography since their expansion complexity is small and their correlation measure of order 2 is large. These facts imply that these sequences are highly predictable despite the fact that they have a large maximum order complexity. Sun and Winterhof (2019) showed that the Thue-Morse sequence along squares keeps a large maximum order complexity. Since, by Christol's theorem, the expansion complexity of this rarefied sequence is no longer bounded, this provides a potentially better candidate for cryptographic applications. Similar results are known for the Rudin-Shapiro sequence and more general pattern sequences. In this paper we generalize these results to any polynomial subsequence (instead of squares) and thereby answer an open problem of Sun and Winterhof. We conclude this paper by some open problems.
We estimate Gowers uniformity norms for some classical automatic sequences, such as the Thue-Morse and Rudin-Shapiro sequences. The methods can also be extended to other automatic sequences. As an application, we asymptotically count arithmetic progressions in the set of integers $\leq N$ where the Thue-Morse (resp. Rudin-Shapiro) sequence takes the value $+1$.
We consider a numeration system in the ring of integers ${\mathcal O}_K$ of a number field, which we assume to be principal. We prove that the property of being a prime in ${\mathcal O}_K$ is decorrelated from two fundamental examples of automatic sequences relative to the chosen numeration system: the Thue-Morse and the Rudin-Shapiro sequences. This is an analogue, in ${\mathcal O}_K$, of results of Mauduit-Rivat which were concerned with the case $K={\mathbb Q}$.
Motivated by the known autocorrelation properties of the Rudin-Shapiro sequence, we study the discrete correlation among infinite sequences over a finite alphabet, where we just take into account whether two symbols are identical. We show by combinatorial means that sequences cannot be "too" different, and by an explicit construction generalizing the Rudin-Shapiro sequence, we show that we can achieve the maximum possible difference.
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…
We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.
We introduce the sequence $(i_n)_{n \geq 0}$ defined by $i_n = (-1)^{inv_2(n)}$, where $inv_2(n)$ denotes the number of inversions (i.e., occurrences of 10 as a scattered subsequence) in the binary representation of n. We show that this sequence has many similarities to the classical Rudin-Shapiro sequence. In particular, if S(N) denotes the N-th partial sum of the sequence $(i_n)_{n \geq 0}$, we show that $S(N) = G(\log_4 N)\sqrt{N}$, where G is a certain function that oscillates periodically between $\sqrt{3}/3$ and $\sqrt{2}$.
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.
We show that the Möbius function is orthogonal to the Thue-Morse sequence $t(n)$ taken along the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ for any $1 < c < 2$. Previously this property was established for the subsequence along the squares $t(n^2)$. These are both examples of Möbius orthogonal sequences with maximum entropy.
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…
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 $α\geq 0$.
In 2009, Grant, Shallit, and Stoll constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefre…
We prove that for $1<c<4/3$ the subsequence of the Thue--Morse sequence $\mathbf t$ indexed by $\lfloor n^c\rfloor$ defines a normal sequence, that is, each finite sequence $(\varepsilon_0,\ldots,\varepsilon_{T-1})\in \{0,1\}^T$ occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf t\left(\lfloor n^c\rfloor\right)$ with asymptotic frequency $2^{-T}$.
We characterize the limiting distribution of Rudin-Shapiro polynomials, showing that, normalized, their values become uniformly distributed in the disc. This resolves conjectures of Saffari and Montgomery. Our proof proceeds by relating the polynomials' distribution to that of a product of weakly dependent random matrices, which we analyze using the representation theory of $SU(2)$. Our approach leads us to a non-commutative analogue of the classical central limit theorem of Salem and Zygmund, which may be of independent interest.
We consider the class of Rudin-Shapiro-like polynomials, whose $L^4$ norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial $f(z)=f_0+f_1 z + \cdots + f_d z^d$ is identified with the sequence $(f_0,f_1,\ldots,f_d)$ of its coefficients. From the $L^4$ norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. In this paper, we study the crosscorrelation properties of pairs of sequences associated to Rudin-Shapiro-like polynomials. We find an explicit formula for the crosscorrelation merit factor. …
Let $q=p^r$ be the power of a prime $p$ and $(β_1,\ldots ,β_r)$ be an ordered basis of $\mathbb{F}_q$ over $\mathbb{F}_p$. For $$ ξ=\sum\limits_{j=1}^r x_jβ_j\in \mathbb{F}_q \quad \mbox{with digits }x_j\in\mathbb{F}_p, $$ we define the Rudin-Shapiro function $R$ on $\mathbb{F}_q$ by $$ R(ξ)=\sum\limits_{i=1}^{r-1} x_ix_{i+1}, \quad ξ\in \mathbb{F}_q. $$ For a non-constant polynomial $f(X)\in \mathbb{F}_q[X]$ and $c\in \mathbb{F}_p$ we study the number of solutions $ξ\in \mathbb{F}_q$ of $R(f(ξ))=c$. If the degree $d$ of $f(X)$ is fixed, $r\ge 6$ and $p\rightarrow \infty$, the number of soluti…
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 …
Borwein and Mossinghoff investigated the Rudin-Shapiro-like sequences, which are infinite families of binary sequences, usually represented as polynomials. Each family of Rudin-Shapiro-like sequences is obtained from a starting sequence (which we call the seed) by a recursive construction that doubles the length of the sequence at each step, and many sequences produced in this manner have exceptionally low aperiodic autocorrelation. Borwein and Mossinghoff showed that the asymptotic autocorrelation merit factor for any such family is at most $3$, and found the seeds of length $40$ or less that…
We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents $P_n(x)/Q_n(x)$ modulo $4$, we give the formal power series expansions (modulo $4$) of these two continued fractions and prove that they are congruent modulo $4$ to algebraic series in $\mathbb{Z}[[x]]$. Therefore, the coefficient sequences of the formal power series expansions are $2$-automatic. Write $Q_{n}(x)=\sum_{i\ge 0}a_{n,i}x^{i}$. Then $(Q_{n}(x))_{n\ge 0}$ defines a two-dimensional coefficient sequence $(a…
In this paper, we generalize Mauduit and Rivat's theorem on the Rudin-Shapiro sequence. Weakening the hypothesis needed in their theorem, we prove a prime number theorem for a large class of functions defined on the digits. Our result covers the case of generalized Rudin-Shapiro sequences as well as bloc-additive sequences on finite and infinite expansions. We also give a partial answer to a question posed by Kalai.
We prove that the Thue--Morse sequence $\mathbf t$ along subsequences indexed by $\lfloor n^c\rfloor$ is normal, where $1<c<3/2$. That is, for $c$ in this range and for each $ω\in\{0,1\}^L$, where $L\geq 1$, the set of occurrences of $ω$ as a subword (contiguous finite subsequence) of the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ has asymptotic density $2^{-L}$. This is an improvement over a recent result by the second author, which handles the case $1<c<4/3$. In particular, this result shows that for $1<c<3/2$ the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ attains both of its…
We show that polynomials associated with automatic sequences satisfy a certain recurrence relation when evaluated at a root of unity, which generalizes a result of Brillhart, Lomont and Morton on the Rudin--Shapiro polynomials. We study the minimal order of such a relation and the integrality of its coefficients.
Integer sequences of the form $\lfloor n^c\rfloor$, where $1<c<2$, can be locally approximated by sequences of the form $\lfloor nα+β\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \[\sum_{n\leq x}φ\left(\lfloor n^c\rfloor\right)-\frac 1c\sum_{n\leq x^c}φ(n)n^{\frac 1c-1},\] which measures the deviation of the mean value of $φ$ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1<c\leq 1.42$ the subsequence of the Thue-Morse sequence indexed by $\lfl…
The $N$th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large $N$th linear complexity. However, in this paper we show that for $q$-automatic sequences over $\mathbb{F}_q$ the converse is not true. We prove that any (not ultimately periodic) $q$-automatic sequence over $\mathbb{F}_q$ has $N$th linear complexity of order of magnitude $N$. For some famous sequences including the Thue--Morse and Rudin--Shapiro sequence we determine the exact values of their $N$th linear complexities. These are non-trivial examples of predictable sequences with…
Many automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, have some desirable features of pseudorandomness such as a large linear complexity and a small well-distribution measure. However, they also have some disastrous properties in view of certain applications. For example, the majority of possible binary patterns never appears in automatic sequences and their correlation measure of order 2 is extremely large. Certain subsequences, such as automatic sequences along squares, may keep the good properties of the original sequence but avoid the bad ones. In this su…
