Showing papers by "Van Vu published in 2005"

Spectral norm of random matrices

Abstract: In this paper, we present a new upper bound for the spectral norm of symmetric random matrices with independent (but not necessarily identical) entries. Our results improve an earlier result of Furedi and Komlos and also correct an incomplete argument in their proof.

On the singularity probability of random Bernoulli matrices

Abstract: Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability 1/2). We show that the probability that $M_n$ is singular is at most $(3/4 +o(1))^n$, improving an earlier estimate of Kahn, Komlos and Szemeredi, as well as earlier work by the authors. The key new ingredient is the applications of Freiman type inverse theorems and other tools from additive combinatorics.

Sharp concentration of random polytopes

Abstract: We prove that key functionals (such as the volume and the number of vertices) of a random polytope is strongly concentrated, using a martingale method. As applications, we derive new estimates for high moments and the speed of convergence of these functionals.

On a question of Erdős and Moser

Abstract: For two finite sets of real numbers A and B , one says that B is sum-free with respect to A if the sum set b + b ' ∣ b , b ' ∈ B , b ≠ b ' is disjoint from A . Forty years ago, Erdőos and Moser posed the following question. Let A be a set of n real numbers. What is the size of the largest subset B of A which is sum-free with respect to A ? In this paper, we show that any set A of n real numbers contains a set B of cardinality at least g ⁡ n ln ⁡ n which is sum-free with respect to A , where g ⁡ n tends to infinity with n . This improves earlier bounds of Klarner, Choi, and Ruzsa and is the first superlogarithmic bound for this problem. Our proof combines tools from graph theory together with several fundamental results in additive number theory such as Freiman's inverse theorem, the Balog-Szemeredi theorem, and Szemeredi's result on long arithmetic progressions. In fact, in order to obtain an explicit bound on g ⁡ n , we use the recent versions of these results, obtained by Chang and by Gowers, where significant quantitative improvements have been achieved.

Long arithmetic progressions in sumsets: Thresholds and bounds

Abstract: l∗A = {a1 + · · ·+ al|ai ∈ Ai, ai = aj} Among the most well-known results in all of mathematics are Vinogradov’s theorem, which says that 3P (P is the set of primes) contains all sufficiently large odd numbers, and Waring’s conjecture (proved by Hilbert, Hardy and Littlewood, Hua, and many others), which asserts that for any given r, there is a number l such that l∗Nr (N denotes the set of rth powers) contains all sufficiently large positive integers (see [29] for an excellent exposition concerning these results) In recent years, a considerable amount of attention has been paid to the study of finite sumsets Given a finite set A and a positive integer l, the natural analogue of Vinogadov-Waring results is to show that under proper conditions, the sumset lA (l∗A) contains a long arithmetic progression Let us assume that A is a subset of the interval [n] = {1, , n}, where n is a large positive integer The concrete problem we would like to address is to estimate the minimum length of the longest arithmetic progression in lA (l∗A) as a function of l, n, and |A| We denote this function by f(|A|, l, n) (f∗(|A|, l, n)), following the notation in [13] Many estimates for f(|A|, l, n) have been discovered by Bourgain, Freiman, Halberstam, Green, Ruzsa, and Sarkozy (see Section 3), but most of these results focus on sets with very high density, namely |A| is close to n Estimating f∗(|A|, l, n) seems much harder, and not much was known prior to our study

Method for designating communication paths in a network

Abstract: A method for designating communication paths in a computer network is provided, in which communication paths are designated for the transmission of data throughout a network. The network may have both recipient computers, which are the intended recipients of the data, and intermediary computers, which are not the intended recipients, but merely relay the data. Each intermediary computer is grouped with the “closest” recipient computer (i.e. the recipient computer with whom it is “least expensive” to communicate). Communication paths between the resulting groups are then identified. A representation of the network is then created. The representation replaces the intermediary computers with the inter-group communication paths, so that the inter-group communication paths appear to pass directly through the locations occupied by the intermediary computers. The created representation is then further processed so that the “least expensive” communication paths may be designated.

Exact k-Wise Intersection Theorems

Abstract: A typical problem in extremal combinatorics is the following. Given a large number n and a set L, find the maximum cardinality of a family of subsets of a ground set of n elements such that the intersection of any two subsets has cardinality in L. We investigate the generalization of this problem, where intersections of more than 2 subsets are considered. In particular, we prove that when k?1 is a power of 2, the size of the extremal k-wise oddtown family is (k?1)(n? 2log2(k?1)). Tight bounds are also found in several other basic cases.

Random symmetric matrices are almost surely non-singular

Abstract: Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that $Q_n$ is non-singular with probability $1-O(n^{-1/8+\delta})$ for any fixed $\delta > 0$. The proof uses a quadratic version of Littlewood-Offord type results concerning the concentration functions of random variables and can be extended for more general models of random matrices.

Discrepancy After Adding A Single Set

Abstract: We show that the hereditary discrepancy of a hypergraph $$ {\user1{\mathcal{F}}} $$ on n points increases by a factor of at most O(log n) when one adds a new edge to $$ {\user1{\mathcal{F}}} $$ .

On random pm 1 matrices: singularity and determinant

Abstract: We proved several results concerning the determinant of a random pm 1 matrix. In particular, we show that with high probability, the determinant has absolute value very close to √n!.

Random Inscribing Polytopes

Abstract: For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem.

Inverse Littlewood-Offord theorems and the condition number of random discrete matrices

Abstract: Consider a random sum $\eta_1 v_1 + ... + \eta_n v_n$, where $\eta_1,...,\eta_n$ are i.i.d. random signs and $v_1,...,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $\P(\eta_1 v_1 + ... + \eta_n v_n = 0)$ subject to various hypotheses on the $v_1,...,v_n$. In this paper we develop an \emph{inverse} Littlewood-Offord theorem (somewhat in the spirit of Freiman's inverse sumset theorem), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the $v_1,...,v_n$ are efficiently contained in an arithmetic progression. As an application we give some new bounds on the distribution of the least singular value of a random Bernoulli matrix, which in turn gives upper tail estimates on the condition number.

Long arithmetic progressions in sumsets: Thresholds and Bounds

Abstract: For a set $A$ of integers, the sumset $lA =A+...+A$ consists of those numbers which can be represented as a sum of $l$ elements of $A$ $$lA =\{a_1+... a_l| a_i \in A_i \}. $$ A closely related and equally interesting notion is that of $l^{\ast}A$, which is the collection of numbers which can be represented as a sum of $l$ different elements of $A$ $$l^{\ast} A =\{a_1+... a_l| a_i \in A_i, a_i eq a_j \}. $$ The goal of this paper is to investigate the structure of $lA$ and $l^{\ast}A$, where $A$ is a subset of $\{1,2, ..., n\}$. As applications, we solve two conjectures by Erdos and Folkman, posed in sixties.

Central limit theorems for random polytopes in a smooth convex set

Abstract: Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove that several key functionals of $K_n$ satisfy the central limit theorem as $n$ tends to infinity.