Related papers: Periodicity of complementing multisets

We prove that if $\max\{|a_0|,|a_1|\}\le 10$ and $\lambda\in\ ]-2,2[$, then the sequence defined by $$ 0 \le a_{n+2} +\lambda a_{n+1}+a_n<1 $$ is periodic.

We define an algorithm which begins with an sequence of sequences, and produces a single sequence, with following property: If at least one of the original sequences has a tail that is periodic, then the output sequence has a periodic tail,…

For a given partition of (1, 2, ..., 2n) into two disjoint subsets A and B with n elements in each, consider the maximum number of times any integer occurs as the difference between an element of A and an element of B. The minimum value of…

A palindromic periodicity is a factor of an infinite word $(ps)^\omega$ where $p$ and $s$ are palindromes and the factor has length at least $|ps|$, for example, $accabaccab$. In this paper we describe several ways in which a palindromic…

For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically,…

We say that an infinite word w is weak abelian periodic if it can be factorized into finite words with the same frequencies of letters. In the paper we study properties of weak abelian periodicity, its relations with balance and frequency.…

The Morse-Hedlund Theorem states that a bi-infinite sequence $\eta$ in a finite alphabet is periodic if and only if there exists $n\in\N$ such that the block complexity function $P_\eta(n)$ satisfies $P_\eta(n)\leq n$. In dimension two,…

Let A be a finite subset of the natural numbers containing 0, and let f(n) denote the number of ways to write n in the form $\sum e_j2^j$, where $\e_j \in A$. We show that there exists a computable T = T(A) so that the sequence (f(n) mod 2)…

We investigate arithmetic properties of the sequence b(n) = B_MN(n) mod M obtained from the base-M to base-N shift map B_MN.We prove that b(n) is ultimately periodic exactly when every prime divisor of M also divides N; in that case we…

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than…

The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal…

Let k be a field of characteristic zero. Given an ordered 3-tuple of positive integers a=(a,b,c) and for j in N, a family of sequences a_j = (j,a+j,a+b+j, a+b+c+j), we consider the collection of monomial curves in A^{4} associated with a_j.…

For a set $A$ of $k$ elements from an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

Let $A, B \subseteq \mathbb{N}$ be two finite sets of natural numbers. We say that $B$ is an additive divisor for $A$ if there exists some $C \subseteq \mathbb{N}$ with $A = B+C$. We prove that among those subsets of $\{0, 1, \ldots, k\}$…

In this paper we study spectral sets which are unions of finitely many intervals in R. We show that any spectrum associated with such a spectral set is periodic, with the period an integral multiple of the measure of the set. As a…

When $p(t)$ is a polynomial of degree $d$, $k$-th column of the Riordan array $\bigl(1/(1 - t^{d+1}), tp(t)\bigr)$ is an eventually periodic sequence with the repeating part beginning at the $1 + (k-1)(d+1)$-st term. The pre-periodic terms…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A =…