173 research outputs found

    Small ball probability, Inverse theorems, and applications

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    Let ξ\xi be a real random variable with mean zero and variance one and A=a1,...,anA={a_1,...,a_n} be a multi-set in Rd\R^d. The random sum SA:=a1ξ1+...+anξnS_A := a_1 \xi_1 + ... + a_n \xi_n where ξi\xi_i are iid copies of ξ\xi is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that SAS_A belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets AA where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page

    Non-abelian Littlewood-Offord inequalities

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    In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.Comment: 14 pages Second version. Dependence of the upper bound on the matrix size in the main results has been remove

    Optimal Inverse Littlewood-Offord theorems

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    Let eta_i be iid Bernoulli random variables, taking values -1,1 with probability 1/2. Given a multiset V of n integers v_1,..., v_n, we define the concentration probability as rho(V) := sup_{x} Pr(v_1 eta_1+...+ v_n eta_n=x). A classical result of Littlewood-Offord and Erdos from the 1940s asserts that if the v_i are non-zero, then rho(V) is O(n^{-1/2}). Since then, many researchers obtained improved bounds by assuming various extra restrictions on V. About 5 years ago, motivated by problems concerning random matrices, Tao and Vu introduced the Inverse Littlewood-Offord problem. In the inverse problem, one would like to give a characterization of the set V, given that rho(V) is relatively large. In this paper, we introduce a new method to attack the inverse problem. As an application, we strengthen a previous result of Tao and Vu, obtaining an optimal characterization for V. This immediately implies several classical theorems, such as those of Sarkozy-Szemeredi and Halasz. The method also applies in the continuous setting and leads to a simple proof for the beta-net theorem of Tao and Vu, which plays a key role in their recent studies of random matrices. All results extend to the general case when V is a subset of an abelian torsion-free group and eta_i are independent variables satisfying some weak conditions

    Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem: a shortened version

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    Let X,X1,…,XnX,X_1,\ldots,X_n be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums ∑k=1nXkak\sum_{k=1}^{n} X_k a_k with respect to the arithmetic structure of coefficients~aka_k in the context of the Littlewood--Offord problem. Concentration results of this type received renewed interest in connection with distributions of singular values of random matrices. Recently, Tao and Vu proposed an Inverse Principle in the Littlewood--Offord problem. We discuss the relations between the Inverse Principle of Tao and Vu as well as that of Nguyen and Vu and a similar principle formulated for sums of arbitrary independent random variables in the work of Arak from the 1980's. This paper is a shortened and edited version of the preprint arXiv:1506.09034. Here we present the results without proofs.Comment: 9 pages. shortened version of arXiv:1506.0903

    Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices

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    Let MnM_n denote a random symmetric nn by nn matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that MnM_n is non-singular with probability 1−O(n−C)1-O(n^{-C}) for any positive constant CC. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.Comment: Some minor corrections in Section 10 of v
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