Article
Bounds on the concentration function in terms of Diophantine approximation
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Comptes Rendus Mathematique (Impact Factor: 0.43). 07/2007; DOI: 10.1016/j.crma.2007.10.006 Source: arXiv

Article: On the LittlewoodOfford problem
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ABSTRACT: The paper deals with studying a connection of the LittlewoodOfford problem with estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of results of Arak (1980) are given. They show a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random vectors for arbitrary distributions of summands.11/2014;  [Show abstract] [Hide abstract]
ABSTRACT: For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator ${S : \ell_p^n \to \mathbb{R}^N}$ such that for every 0 < r < p < 2 with r ≤ 1, the operator ${S_r = S : \ell_p^n \to \ell_r^N}$ satisfies with overwhelming probability that ${\S_r\ \, \(S_r)_{ {\rm Im}\, S}^{1}\ \le C(p,r)^{n/(Nn)}}$ , where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.Mathematische Annalen 01/2011; 350(4). · 1.20 Impact Factor 
Article: Random embedding of ℓ p n into ℓ r N
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ABSTRACT: The paper is devoted to “proportional” embeddings of ℓ p n into ℓ r N . A typical result in this direction is the B. S. Kashin [Math. USSR, Izv. 11, 317–333 (1977; Zbl 0378.46027)] theorem: For any η>0, for any n, ℓ 2 n ↪ cℓ 1 N , where N=(1+η)n and the constant of isomorphism c depends only on η. All results of this sort are random: one constructs a random operator from ℓ p n into ℓ r N and proves that with a positive probability this operator has “nice” constant of isomorphism. The authors give, for any 0<p<2 and any natural numbers n<N, an explicit definition of a random operator S:ℓ p n →ℝ N with the following property. For every 0<r<p, r≤1, the operator S r =S:ℓ p n →ℓ r N satisfies with overwhelming probability that ∥S r ∥∥S r 1 ∥≤c n/(Nn) , where c>0 depends only on p and r. The authors note that these operators S r have already been defined in [G. Pisier, Trans. Am. Math. Soc., 276, 201–211 (1983; Zbl 0509.46016)] for the almost isometric result.Mathematische Annalen 01/2011; 350(4). · 1.20 Impact Factor
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