Related papers: On the Limiting Density of a gcd Map

We show that the symmetry of \[f\left(a,b\right)=\frac{\operatorname{gcd}\left(ab,a+b\right)}{\operatorname{gcd}\left(a,b\right)}\] stems from an $\operatorname{SL}_2\left(\mathbb{Z}\right)$ action on primitive pairs and that all solutions…

Let $\mathbf{D}=(D_{n})_{n\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\mathscr{A}_{E,k}=\{n\geq 1 : \gcd(n,D_{n})=k\}$. We give an explicit structural description of…

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if $f$ is a probability density on $\R^n$ of the form $f(x)=\prod_{i=1}^n f_i(x_i)$, where each $f_i$ is a density on $\R$, say…

We show the following result: Assume B is an infinite Boolean Algebra and lambda=d(B). Then s(B*B)$, i.e. s(uf(B)xuf(B))>= lambda$ (if lambda limit - obtained)

We prove that for all constants $a\in\N$, $b\in\Z$, $c,d\in\R$, $c\neq 0$, the fractions $\phi(an+b)/(cn+d)$ lie dense in the interval $]0,D]$ (respectively $[D,0[$ if $c<0$), where $D=a\phi(\gcd(a,b))/(c\gcd(a,b))$. This interval is the…

Fix integers $b>a\geq1$ with $g:=\gcd(a,b)$. A set $S\subseteq\mathbb{N}$ is \emph{$\{a,b\}$-multiplicative} if $ax\neq by$ for all $x,y\in S$. For all $n$, we determine an $\{a,b\}$-multiplicative set with maximum cardinality in $[n]$, and…

We define the limiting density of a minor-closed family of simple graphs F to be the smallest number k such that every n-vertex graph in F has at most kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting densities.…

In this paper we study the density in the real line of oscillating sequences of the form $$ (g(k)\cdot F(k\alpha))_{k \in \mathbb{N}} ,$$ where $g$ is a positive increasing function and $F$ a real continuous 1-periodic function. This…

For each positive integer $k$, let $\mathscr{A}_k$ be the set of all positive integers $n$ such that $\gcd(n, F_n) = k$, where $F_n$ denotes the $n$th Fibonacci number. We prove that the asymptotic density of $\mathscr{A}_k$ exists and is…

We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural ``limit object'', namely a symmetric measurable 2-variable…

Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p=a(mod f) and g is a primitive root modulo p has a…

Packings of equal disks in the plane are known to have density at most $\pi/\sqrt{12}$, although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a…

Let $a_0=b_0=0$ and $0<a_1\leq b_1<a_2\leq b_2<\ldots\leq b_{n}$ be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in…

Let $F$ and $G$ be integer polynomials where $F$ has degree at least $2$. Define the sequence $(a_n)$ by $a_n=F(a_{n-1})$ for all $n\ge 1$ and $a_0=0.$ Let $\mathscr{B}_{F,\,G,\,k}$ be the set of all positive integers $n$ such that $k\mid…

We consider the density at a point z = x + i y of critical percolation clusters that touch the left [P_L(z)], right [P_R(z)], or both [P_{LR}(z)] sides of a rectangular system, with open boundary conditions on the top and bottom. The ratio…

In Theorem 1 of Acta Arith. 99 (2001), 321-329, Cobeli and Zaharescu give a result about the distribution of the ${\bf F}_p$-points on an affine curve. An easy corollary to their theorem is that the set $\bigcup_p \lbrace (\frac{x_1}{p},…

We study densities of $p$-adically bounded primes for hypergeometric series in two cases: the case of generalized hypergeometric series with rational parameters, and the case of $_2F_1$ with parameters in a quadratic extension of the…

Let $f(t_1,\ldots,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least…

We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such…

How small can a set of vertices in the $n$-dimensional hypercube $Q_n$ be if it meets every copy of $Q_d$? The asymptotic density of such a set (for $d$ fixed and $n$ large) is denoted by $\gamma_d$. It is easy to see that $\gamma_d \leq…