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

Suppose l=2m+1, m>0. We introduce m "theta-series", [1],...,[m], in Z/2[[x]]. It has been conjectured that the n for which the coefficient of x^n in 1/[i] is 1 form a set of density 0. This is probably always false, but in certain cases,…

We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some…

Our main result states that when A, B, C are subsets of Z/NZ of respective densities \alpha,\beta,\gamma, the sumset A + B + C contains an arithmetic progression of length at least e^{c(\log N)^c} for densities \alpha > (\log N)^{-2 +…

We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerden's proof for the existence of arbitrary long…

A novel generalization of the Prouhet-Thue-Morse sequence to binary $\pm 1$-weight sequences is presented. Derived from Rademacher functions, these weight sequences are shown to satisfy interesting orthogonality and recurrence relations. In…

This short note reports a master theorem on tight asymptotic solutions to divide-and-conquer recurrences with more than one recursive term: for example, T(n) = 1/4 T(n/16) + 1/3 T(3n/5) + 4 T(n/100) + 10 T(n/300) + n^2.

We show that the equation $\lambda_1 n_1^2 + ... + \lambda_s n_s^2 = 0$ admits non-trivial solutions in any subset of $[N]$ of density $(\log N)^{-c_s}$, provided that $s \geq 7$ and the coefficients $\lambda_i$ sum to zero and satisfy…

We generalize the construction of Doppler-tolerant Golay complementary waveforms by Pezeshki-Calderbank-Moran-Howard to complementary code sets having more than two codes. This is accomplished by exploiting number-theoretic results…

Let $k\in \mathbb{N}$ and $s\geq k(\log k+3.20032)$. Let $\mathbb{N}_{0}^{k}$ be the set of $k$-th powers of nonnegative integers. Assume that $\psi$ is an increasing function tending to infinity with $\psi(x)=o(\log x)$ and satifying some…

Let $b \ge 2$ be an integer. We prove that the $b$-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms.…

Given a countable set X (usually taken to be the natural numbers or integers), an infinite permutation, \pi, of X is a linear ordering of X. This paper investigates the combinatorial complexity of infinite permutations on the natural…

We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $\log$-uniformly continuous normal order is close to a function of the form $n\mapsto n^c$. As an application…

We solve an asymptotic problem in the geometry of numbers, where we count the number of singular $n\times n$ matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by…

This paper studies non-autonomous Lyness type recurrences of the form $x_{n+2}=(a_n+x_{n+1})/x_{n}$, where $\{a_n\}$ is a $k$-periodic sequence of positive numbers with primitive period $k$. We show that for the cases $k\in\{1,2,3,6\}$ the…

Let $f(n)$ be a Steinhaus random multiplicative function. Let $A\subset [1, N]$ be a finite set of integers. We show that \[\frac{1}{\sqrt{|A|}} \sum_{n\in A} f(n) \xrightarrow[]{d} \mathcal{CN}(0,1)\] forces that $|A|=o(N)$. We prove that…

We describe under a variety of conditions abelian subgroups of the automorphism group A of the regular n-ary tree T which are normalized by the n-ary adding machine t=(e,...,e,t)s where s is the n-cycle (0,1,...,n-1). As an application, for…

We prove that Thue-Morse constant $\tau_{TM}=0.01101001..._2$ is not a badly approximable number. Moreover, we prove that $\tau_{TM}(a)=0.01101001..._a$ is not badly approximable for every integer base $a\geq 2$ such that $a$ is not…

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum…

Let $e_v(n)$ denote the number of occurrences of a fixed pattern $v$ in the binary expansion of $n \in \mathbb{N}$. In this paper we study monochromatic arithmetic progressions in the class of binary words $(e_v(n) \bmod{2})_{n \geq 0}$,…

Denote by $\tau$ k (n), $\omega$(n) and $\mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let…