Related papers: On the additive complexity of a Thue-Morse like se…

For $q \geq 2$, $n \in \mathbb{N}$, let $s_{q}(n)$ denote the sum of the digits of $n$ written in base $q$. Spiegelhofer (2020) proved that the Thue--Morse sequence has level of distribution $1$, improving on a former result of Fouvry and…

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…

Let $p$ be a prime number and consider a $p$-automatic sequence ${\bf u}=(u_{n})_{n\in\N}$ and its generating function $U(X)=\sum_{n=0}^{\infty}u_{n}X^{n}\in\mathbb{F}_{p}[[X]]$. Moreover, let us suppose that $u_{0}=0$ and $u_{1}\neq 0$ and…

In this paper, we study the $2$-binomial complexity $b_{\mathbf{t}_{m},2}(n)$ of the generalized Thue-Morse words $\mathbf{t}_{m}$ for every integer $m\geq 3$. We obtain the exact value of $b_{\mathbf{t}_{m},2}(n)$ for every integer $n\geq…

Let $f: \mathbb{N} \to \mathbb{C}$ be a multiplicative function for which $$ \sum_{p : \, |f(p)| \neq 1} \frac{1}{p} = \infty. $$ We show under this condition alone that for any integer $h \neq 0$ the set $$ \{n \in \mathbb{N} : f(n) =…

Let $(u_n)_{n\ge 0}$ denote the Thue-Morse sequence with values $\pm 1$. The Woods-Robbins identity below and several of its generalisations are well-known in the literature…

In this paper we study some additive properties of subsets of the set $\nats$ of positive integers: A subset $A$ of $\nats$ is called {\it $k$-summable} (where $k\in\ben$) if $A$ contains $\textstyle \big{\sum_{n\in F}x_n | \emp\neq…

We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a $2$-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every…

Let, for r>=2, (m_r(n)),n>=0, be Moser sequence such that every nonnegative integer is the unique sum of the form s_k+rs_l. In this article we give an explicit decomposition formulas of such form and an unexpectedly simple recursion…

Letting $w$ denote a finite, nonempty word, let $\text{red}(w)$ denote the word obtained from $w$ by replacing every subword $s$ of $w$ of the form $cc \cdots c$ for a given character $c$ (such that there is no character immediately to the…

Given a sequence $\mathscr{A}=\{a_0<a_1<a_2\ldots\}\subseteq \mathbb{N}$, let $r_{\mathscr{A},h}(n)$ denote the number of ways $n$ can be written as the sum of $h$ elements of $\mathscr{A}$. Fixing $h\geq 2$, we show that if $f$ is a…

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…

Here is a sample of the results proved in this paper: Let $f:{\bf R}\to {\bf R}$ be a continuous function, let $\rho>0$ and let $\omega:[0,\rho[\to [0,+\infty[$ be a continuous increasing function such that $\lim_{\xi\to…

Using known results about their mod-2 cohomology ring, we prove that the topological complexity of the space of isometry classes of n-gons in the plane with one side of length r and all others of length 1 equals either 2n-5 or 2n-6,…

Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1…

We study the behavior of a polynomial sequence which is defined by iterating a polynomial pair under Thue-Morse dynamic. We show that in suitable sense, the sequence will behave like $\{2\cos 2^nx: n\ge 1\}$. Basing on this property we can…

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n \alpha)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $\alpha$. Another…

This paper examines Automatic Complexity, a complexity notion introduced by Shallit and Wang in 2001. We demonstrate that there exists a normal sequence $T$ such that $I(T) = 0$ and $S(T) \leq 1/2$, where $I(T)$ and $S(T)$ are the lower and…

Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer $n\geq 0$…

A sequence of positive integers $(a_1,a_2,\ldots,a_k)$ is called $\ell$-additive if $a_1+a_2+\cdots+a_k=\ell a_1$ or $\ell a_k$. In this paper, we prove that for all $k\geq3$, if $n$ is sufficiently large, then every permutation of…