我们研究各种实值多项式$p$和衰减率$\varepsilon$的形式$A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$的集合。特别是，我们问，当这样的集合是基地的有限阶的正整数。我们表明，一般来说，$A$是一个基础的顺序$2$时，$\operatorname{deg} p \geq 3$，但不是当$\operatorname{deg} p = 2$，虽然然后$A + A$仍然有渐近密度$1$。
英文摘要：
We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets are bases of finite order for the positive integers. We show that generically, $A$ is a basis of order $2$ when $\operatorname{deg} p \geq 3$, but not when $\operatorname{deg} p = 2$, although then $A + A$ still has asymptotic density $1$.
We propose a new definition of effective formulas for problems in enumerative combinatorics. We outline the proof of the fact that every linear recurrence sequence of integers has such a formula. It follows from a lower bound that can be deduced from the Skolem-Mahler-Lech theorem and the Subspace Theorem. We will give details of this deduction that is due to P. Corvaja in the full version of this extended abstract.
Let $\{U_n\}_{n \geqslant 0}$ and $\{G_m\}_{m \geqslant 0}$ be two linear recurrence sequences defined over the integers. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as differences $ U_n - G_m$, when $x$ goes to infinity. In particular, the density of such integers is $0$.
Let $\{U_n\}_{n \geq 0}$ and $\{V_m\}_{m \geq 0}$ be two linear recurrence sequences. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as differences $ U_n - V_m$. In particular, the density of such integers is $0$.
It is commonly known that $ζ(2k) = q_{k}\frac{ζ(2k + 2)}{π^2}$ with known rational numbers $q_{k}$. In this work we construct recurrence relations of the form $\sum_{k = 1}^{\infty}r_{k}\frac{ζ(2k + 1)}{π^{2k}} = 0$ and show that series representations for the coefficients $r_{k} \in \mathbb{R}$ can be computed explicitly.
Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. These decompositions are generalizations of base $B$ decompositions. In this and the follow-up paper, we provide two approaches to investigate linear recurrences with leading coefficient zero…
Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. In this and the preceding paper we provide two approaches to investigate linear recurrences with leading coefficient zero, followed by non-negative integer coefficients, with differences betw…
For positive integers $α$ and $β$, we define an $(α,β)$-walk to be any sequence of positive integers satisfying $w_{k+2}=αw_{k+1}+βw_k$. We say that an $(α,β)$-walk is $n$-slow if $w_s=n$ with $s$ as large as possible. Slow $(1,1)$-walks have been investigated by several authors. In this paper we consider $(α,β)$-walks for arbitrary positive $α,β$. We derive a characterization theorem for these walks, and with this we prove several results concerning the total number of $n$-slow walks for a given $n$. In addition to this, we study the slowest $n$-slow walk for a given $n$ amongst all possible …
We prove that there is a set of integers $A$ having positive upper Banach density whose difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any integer, answering a question asked by Bergelson, Hegyvári, Ruzsa, and the author, in various combinations. In the language of dynamical systems, this result shows that there is a set of integers $S$ which is dense in the Bohr topology of $\mathbb Z$ and which is not a set of measurable recurrence. Our proof yields the following stronger result: if $S\subseteq \mathbb Z$ is dense in the Bohr topology of $\mathbb Z$, then ther…
For each positive integer $N$, define $$S'_N \ =\ \{1 < d < \sqrt{N}: d|N\}\mbox{ and }L'_N \ =\ \{\sqrt{N} < d < N : d|N\}.$$ Recently, Chentouf characterized all positive integers $N$ such that the set of small divisors $\{d\le \sqrt{N}: d|N\}$ satisfies a linear recurrence of order at most two. We nontrivially extend the result by excluding the trivial divisor $1$ from consideration, which dramatically increases the analysis complexity. Our first result characterizes all positive integers $N$ such that $S'_N$ satisfies a linear recurrence of order at most two. Moreover, our second result ch…
Let $S= \{ p_1, \ldots, p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_{n})_{n \geq 0}$ be a linear recurrence sequence of integers of order $r$. For any positive integer $k,$ we define $(U_j^{(k)})_{j\geq 1}$ an increasing sequence composed of integers of the form $U_{n_k} +\cdots + U_{n_1}, \ n_k>\cdots >n_1$. Under certain assumptions, we prove that for any $ε>0,$ there exists an integer $n_{0}$ such that $[U_j^{(k)}]_S < \left(U_j^{(k)}\right)^ε,$ for $ j > n_0,$ where $[m]_S$ denote the $S$-part of the positive integer $m$. On further assumptions on $(U_{n})_{n \geq 0…
We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex plane, giving explicit formulas for its pole set and residues, as well as for its finite values at negative integers, which are shown to be rational numbers. To illustrate the results, we focus on some concrete examples which have also been studied previously by other authors.
Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree $d$ whose Galois group is $S_d$. Let $(a_n)$ be a linearly recuresive sequence of integers which has $P$ as its characteristic polynomial. We prove, under the generalized Riemann hypothesis, that the lower density of the set of primes which divide at least one element of the sequence $(a_n)$ is positive.
For every nonconstant monic polynomial $g \in \mathbb{Z}[X]$, let $\mathfrak{M}(g)$ be the set of positive integers $m$ for which there exist an integer linear recurrence $(s_n)_{n \geq 0}$ having characteristic polynomial $g$ and a positive integer $M$ such that $(s_n)_{n \geq 0}$ has exactly $m$ distinct residues modulo $M$. Dubickas and Novikas proved that $\mathfrak{M}(X^2 - X - 1) = \mathbb{N}$. We study $\mathfrak{M}(g)$ in the case in which $g$ is divisible by a monic quadratic polynomial $f \in \mathbb{Z}[X]$ with roots $α,β$ such that $αβ= \pm 1$ and $α/ β$ is not a root of unity. We …
Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in terms of linear recurrence sequence $(U_n)_{n\geq 0}$ under certain weak assumptions has been studied. However, there is an error in their proof. In this paper, we generalize their work, as well as our result fixes their error. In particular, for a given polynomial $Q(x) \in \mathbb{Z}[x]$ we consider the Diophantine equation $U_{n_1} + \cdots + U_{n_k} = \ell…
Given a sequence of distinct positive integers $w_0 , w_1, w_2, \ldots$ and any positive integer $n$, we define the discriminator function $\mathcal{D}_{\bf w}(n)$ to be the smallest positive integer $m$ such that $w_0,\ldots, w_{n-1}$ are pairwise incongruent modulo $m$. In this paper, we classify all binary recurrent sequences $\{w_n\}_{n\geq 0}$ consisting of different integer terms such that $\mathcal{D}_{\bf w}(2^e)=2^e$ for every $e\geq 1.$ For all of these sequences it is expected that one can actually give a fairly simple description of $\mathcal{D}_{\bf w}(n)$ for every $n\ge 1.$ For …
There is a growing literature on sums of reciprocals of polynomial functions of recurrence relations with constant coefficients and fixed depth, such as Fibonacci and Tribonacci numbers, products of such numbers, and balancing numbers (numbers $n$ such that the sum of the integers less than $n$ equals the sum of the $r$ integers immediately after, for some $r$ which is called the balancer of $n$; If $n$ is included in the summation, we have the cobalancing numbers, and $r$ is called the cobalancer of $n$). We generalize previous work to reciprocal sums of depth two recurrence sequences with ar…
Given a positive integer $n$, the small divisors of $n$ are defined as the positive divisors that do not exceed $\sqrt{n}.$ Ianucci previously classified all $n$ for which the small divisors of $n$ form an arithmetic progression. In this paper, we classify all $n$ for which the small divisors of $n$ form a linear recurrence of order at most two.
In this paper, we present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let $P,Q,R_0$, and $R_1$ be fixed integers and let $\boldsymbol{R}=\left(R_n\right)_{n}$ be the recurrence sequence defined by $R_{n+2}=PR_{n+1}-QR_{n}$ $(\forall n\geq 0)$. Under some conditions on the parameters, we determine a rational nontrivial divisor for $L_{k,n}:=\mathrm{lcm}\left(R_k,R_{k+1},\dots,R_n\right)$, for all positive integers $n$ and $k$, such that $n\geq k$. As consequences, we derive nontrivial effective lower bounds for $L_{k,n}$ and we estab…
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 set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_p$', \emph{Bull. Aust. Math. Soc.} \textbf{96} (2017), 24-29], Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In this article, we find a sufficient condition for denseness of the quotient set of the $k$th-order linea…
Multiplicative dependence in linear recurrence sequences
University of Debrecen（德布勒森大学）；Count István Tisza Foundation, Institute of Mathematics, University of Debrecen（蒂萨·伊斯特万伯爵基金会，德布勒森大学数学研究所）；HUN-REN-DE Equations, Functions, Curves and their Applications Research Group（匈牙利研究网络-德布勒森方程、函数、曲线及其应用研究组）；University of New South Wales（新南威尔士大学）
Consider a sequence of positive integers of the form $ca^n-d$, $n\geq 1$, where $a, c$ and $d$ are positive integers, $a>1$. For each $n\geq 1$, let $S_n$ be the submonoid of $\mathbb N$ generated by $\mathbf s_j=ca^{n+j}-d$, with $j\in\mathbb N$. We obtain a numerical semigroup $(1/e)S_n$ by dividing every element of $S_n$ by $e=\gcd(S_n)$. We characterize the embedding dimension of $S_n$ and describe a method to find the minimal generating set of $S_n$. We also show how to find the maximum element of the Apéry set ${\rm Ap}(S_n, \mathbf s_0)$, characterize the elements of ${\rm Ap}(S_n, \mat…
Let $n$ and $n'$ be positive integers such that $n-n'\in \{0,1\}$. Let $F$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $K_n$ and $K_{n'}$ be maximal compact subgroups of $\mathrm{GL}(n,F)$ and $\mathrm{GL}(n',F)$, respectively. We give the explicit descriptions of archimedean Rankin--Selberg integrals at the minimal $K_n$- and $K_{n'}$-types for pairs of principal series representations of $\mathrm{GL}(n,F)$ and $\mathrm{GL}(n',F)$, using their recurrence relations. Our results for $F=\mathbb{C}$ can be applied to the arithmetic study of critical values of automorphic $L$-functions.
Linear recurrence relations for binomial coefficients modulo a prime
研究了当正整数k为固定值时，二项式系数序列\binom{k}{i}模素数p在有限范围0 \le i \le k中满足（正）次h的线性递归关系.特别地，我们证明了，如果2 h \le k<p-h，这不会发生。这个假设可以减弱为2 h \le k<p，如果我们假设，此外，关系的特征多项式没有-1作为根。我们应用我们的结果来恢复一个已知的有限域有限域上的费马曲线的点的数目的界限。
For every positive integer $n$ and every $δ\in [0,1]$, let $B(n, δ)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $δ$. Moreover, let $(u_k)_{k \geq 0}$ be an integer sequence satisfying $u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$, for every integer $k \geq 2$, where $u_0 = 0$, $u_1 \neq 0$, and $a_1, a_2$ are fixed nonzero integers; and let $α$ and $β$, with $|α| \geq |β|$, be the two roots of the polynomial $X^2 - a_1 X - a_2$. Also, assume that $α/ β$ is not a root of …
