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: Small ball estimates for quasinorms
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ABSTRACT: This note contains two types of small ball estimates for random vectors in finite dimensional spaces equipped with a quasinorm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain LittlewoodOfford type estimates for quasinorms. This generalizes a result which was previously obtained by Friedland and Sodin and by Rudelson and Vershynin.10/2014; 
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; 
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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