Related papers: Linear Recurrences of Generalized Schreier Sets Re…

A Schreier set $S$ is a subset of the natural numbers with $\min S\ge |S|$. It has been known that the sequence $(a_{1,n})$, where $$a_{1,n}\ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S \ge |S|\}|,$$ is the Fibonacci…

A nonempty set $F$ is Schreier if $\min F\ge |F|$. Bird observed that counting Schreier sets in a certain way produces the Fibonacci sequence. Since then, various connections between variants of Schreier sets and well-known sequences have…

A set $A$ of positive integers is said to be Schreier if either $A = \emptyset$ or $\min A\ge |A|$. We give a bijective map to prove the recurrence of the sequence $(|\mathcal{K}_{n, p, q}|)_{n=1}^\infty$ (for fixed $p\ge 1$ and $q\ge 2$),…

A subset of positive integers $F$ is a Schreier set if it is non-empty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of $F$). For each positive integer $k$, we define $k\mathcal{S}$ as the collection of all the unions of at most…

Recently, a relation between Schreier-type sets and Tur\'{a}n graphs was discovered. In this note, we give a combinatorial proof and obtain a generalization of the relation. Specifically, for $p, q\ge 1$, let $$\mathcal{A}_q :=…

Let $\mathbb{F}_q$ be the field with $q$ elements and of characteristic $p$. For $a\in\mathbb{F}_p$ consider the set \begin{equation*} S_a(n)=\{f\in\mathbb{F}_q[x]\mid\operatorname{deg}(f)=n,~f\text{ irreducible, monic and}…

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$…

Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient…

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…

Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017,…

We prove that the sets $\{n \in N: n$ satisfies formula (1)$\}$ and $\{n \in N: n$ does not satisfy formula (2)$\}$ are not recursively enumerable. We prove that these sets are co-recursively enumerable. $(1)~\exists p,q \in…

Let $(X, \mathcal{B},\mu,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $\epsilon>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \left\{ n\in\mathbb{N}\colon\mu(A\cap…

An infinite $(p,q)$-theorem, or an $(\aleph_0,q)$-theorem, involving two families $\mathcal{F}$ and $\mathcal{G}$ of sets, states that if in every infinite subset of $\mathcal{F}$, there are $q$ sets that are intersected by some set in…

Let $(a_n)_{n=0}^\infty$ be a second-order linear recurrence sequence with constant coefficients over the field of $p$-adic numbers $\mathbb{Q}_p$. We study the set of limit points of the sequence of consecutive ratios…

In a recent article, Donoso, Le, Moreira and Sun studied sets of recurrence for actions of the multiplicative semigroup $(\mathbb{N}, \times)$ and provided some sufficient conditions for sets of the form $S=\{(an+b)/(cn+d) \colon n \in…

In this paper, we study the three-term nested recurrence relation $B(n)=B(n-B(n-1))+B(n-B(n-2))+B(n-B(n-3))$ subject to initial conditions where the first $N$ terms are the integers $1$ through $N$. This recurrence is the three-term analog…

Let $S = \{q_1, \ldots , q_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = q_1^{r_1} \ldots q_s^{r_s} M$, where $r_1, \ldots , r_s$ are non-negative integers and $M$ is an integer relatively…

Fathi and Pageault have recently shown a connection between Auslander's generalized recurrent set $GR(f)$ and Easton's strong chain recurrent set. We study $GR(f)$ by examining that connection in more detail, as well as connections with…

Recently, motivated by the study of vectorized stream cipher systems, the joint linear complexity and joint minimal polynomial of multisequences have been investigated. Let S be a linear recurring sequence over finite field F_{q^m} with…

Let $X = [0,1]$, and let $T:X\to X$ be an expanding piecewise linear map sending each interval of linearity to $[0,1]$. For $\psi:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function \[…