Related papers: Limit points of Nathanson's Lambda sequences

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\lambda,\mu$. For any $\lambda>0$ we consider the percolation…

Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \to \infty$, $M_n- {3 \over 2} \log n$ is tight (see [1][9][2]). We establish here a law of iterated logarithm for the…

Let $A$ be an $AH$ algebra, that is, $A$ is the inductive limit $C^{*}$-algebra of $$A_{1}\xrightarrow{\phi_{1,2}}A_{2}\xrightarrow{\phi_{2,3}}A_{3}\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots$$ with…

For any $n$-tuple $(\alpha_1,...,\alpha_n)$ of linearly independent vectors in Hilbert space $H$, we construct a unique orthonormal basis $(\epsilon_1,...,\epsilon_n)$ of $span\{\alpha_1,...,\alpha_n\}$ satisfying:…

For a given weighted mean $\mathscr{M}$ defined on a subinterval of $\mathbb{R}_+$ and a sequence of weights $\lambda=(\lambda_n)_{n=1}^\infty$ we define a Hardy constant $\mathscr H(\lambda)$ as the smallest extended real number such that…

Let \[Y_j=f_*(X_j)+\xi_j,\qquad j=1,...,n,\] where $X,X_1,...,X_n$ are i.i.d. random variables in a measurable space $(S,\mathcal{A})$ with distribution $\Pi$ and $\xi,\xi_1,... ,\xi_n$ are i.i.d. random variables with ${\mathbb{E}}\xi=0$…

Given a subset $\Lambda$ of $\mathbb Z_+:=\{0,1,2,\dots\}$, let $H^\infty(\Lambda)$ denote the space of bounded analytic functions $f$ on the unit disk whose coefficients $\widehat f(k)$ vanish for $k\notin\Lambda$. Assuming that either…

Erd\H os and R\'{e}nyi claimed and Vu proved that for all $h \ge 2$ and for all $\epsilon > 0$, there exists $g = g_h(\epsilon)$ and a sequence of integers $A$ such that the number of ordered representations of any number as a sum of $h$…

Let G be an arbitrary Abelian group and let A be a finite subset of G. A has small additive doubling if |A+A| < K|A| for some K>0. These sets were studied in papers of G.A. Freiman, Y. Bilu, I. Ruzsa, M.C.--Chang, B. Green and T.Tao. In the…

For a real $N\ge 1$ and a vector $\xi =(1,\xi_1,...,\xi_n)$ define a matrix $$ {\cal A} (\xi, N) = ({array}{ccccc} N^{-1} & 0& 0& ... &0 \cr N^{\frac{1}{n}} \xi_1 & -N^{\frac{1}{n}} & 0&... & 0 \cr N^{\frac{1}{n}} \xi_2 &0& -N^{\frac{1}{n}}…

The study of the well-known partition function $p(n)$ counting the number of solutions to $n = a_{1} + \dots + a_{\ell}$ with integers $1 \leq a_{1} \leq \dots \leq a_{\ell}$ has a long history in combinatorics. In this paper, we study a…

Let $A_n$ be an $n\times n$ random symmetric matrix with $(A_{ij})_{i< j}$ i.i.d. mean $0$, variance 1, following a subGaussian distribution and diagonal elements i.i.d. following a subGaussian distribution with a fixed variance. We…

Let $\mathcal{S}$ be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and $\mathcal{T}$ to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such…

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such…

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of…

Let $G$ be any group and $A$ be an arbitrary subset of $G$ (not necessarily symmetric and not necessarily containing the identity). The $h$-fold product set of $A$ is defined as $$A^{h} :=\lbrace a_{1}.a_{2}...a_{h} : a_{1},\ldots,a_n \in A…

Let $\mathbf{P} \subset [H_0,H]$ be a set of primes, where $\log H_0 \geq (\log H)^{2/3 + \epsilon}$. Let $\mathscr{L} = \sum_{p \in \mathbf{P}} 1/p$. Let $N$ be such that $\log H \leq (\log N)^{1/2-\epsilon}$. We show there exists a subset…

We show that for any finite set $P$ of points in the plane and $\epsilon>0$ there exist $\displaystyle O\left(\frac{1}{\epsilon^{3/2+\gamma}}\right)$ points in ${\mathbb{R}}^2$, for arbitrary small $\gamma>0$, that pierce every convex set…

In this paper, we prove the following result: {quote} Let $\A$ be an infinite set of positive integers. For all positive integer $n$, let $\tau_n$ denote the smallest element of $\A$ which does not divide $n$. Then we have $$\lim_{N \to +…

We show that for any sufficiently large integer $Q$ and a real $0\leq\lambda\leq\frac34$ there exists a value $c(n,f,J)>0$ such that all strips $L(Q,\lambda)=\{(x,y):|y-f(x)|<Q^{-\lambda}, x\in J=[a,b]\}$ contain at least $c(n, f,…