Searcharxiv⌕ Search

arXiv subjects

Omer Friedland

Publications and source records attributed to Omer Friedland.

29 records · Page 2Linked to original sources
arXiv2015

Inverse Littlewood-Offord problems for Quasi-Norms

Given a star-shaped domain $K\subseteq \mathbb R^d$, $n$ vectors $v_1,\dots,v_n \in \mathbb R^d$, a number $R>0$, and i.i.d. random variables $η_1,\dots,η_n$, we study the geometric and arithmetic structure of the set of vectors $V = \{v_1,\dots,v_n\}$ under the assumption that the small ball probability \[\sup_{x\in \mathbb R^d}~\mathbb P\Bigg(\sum_{j=1}^nη_jv_j\in x+RK\Bigg)\] does not decay too fast as $n\to \infty$. This generalises the case where $K$ is the Euclidean ball, which was previously studied by Nguyen-Vu and Tao-Vu.

math.PR↗
arXiv2015

Small ball estimates for quasi-norms

This note contains two types of small ball estimates for random vectors in finite dimensional spaces equipped with a quasi-norm. 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 Littlewood-Offord type estimates for quasi-norms. This generalizes a result which was previously obtained by Friedland and Sodin and by Rudelson and Vershynin.

math.PR↗
arXiv2015

$(s,p)$-Valent Functions

We introduce the notion of $(\mathcal F,p)$-valent functions. We concentrate in our investigation on the case, where $\mathcal F$ is the class of polynomials of degree at most $s$. These functions, which we call $(s,p)$-valent functions, provide a natural generalization of $p$-valent functions (see~\cite{Ha}). We provide a rather accurate characterizing of $(s,p)$-valent functions in terms of their Taylor coefficients, through "Taylor domination", and through linear non-stationary recurrences with uniformly bounded coefficients. We prove a "distortion theorem" for such functions, comparing them with polynomials sharing their zeroes, and obtain an essentially sharp Remez-type inequality in the spirit of~\cite{Y3} for complex polynomials of one variable. Finally, based on these results, we present a Remez-type inequality for $(s,p)$-valent functions.

math.CA↗
arXiv2013

An observation on the Turán-Nazarov inequality

The main observation of this note is that the Lebesgue measure $μ$ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant $ω\ge μ$, which can be effectively estimated in terms of the metric entropy of a set, and may be nonzero for discrete and even finite sets. While the frequencies (the imaginary parts of the exponents) do not enter in the original Turán-Nazarov inequality, they necessarily enter the definition of $ω$.

math.FA↗
arXiv2013

Vitushkin-type theorems

It is shown that for a subset $A\subset\R^n$ that has the global Gabrielov property, a Vitushkin-type estimate holds. Concrete examples are given for sub-level sets of certain classes of functions.

math.CA↗
arXiv2013

A simple observation on random matrices with continuous diagonal entries

Let $T$ be an $n\times n$ random matrix, such that each diagonal entry $T_{i,i}$ is a continuous random variable, independent from all the other entries of $T$. Then for every $n\times n$ matrix $A$ and every $t\ge0$ $$ \p\Big[|\det(A+T)|^{1/n}\le t\Big]\le2bnt, $$ where $b>0$ is a uniform upper bound on the densities of $T_{i,i}$.

math.PR↗
arXiv2011

Sparsity and non-Euclidean embeddings

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of $\ell_p^n$, $p > 0$, into various type of Banach or quasi-Banach spaces. In particular, for $0 0$.

math.FA↗
arXiv2007

Kahane-Khinchin type Averages

We prove a Kahane-Khinchin type result with a few random vectors, which are distributed independently with respect to an arbitrary log-concave probability measure on $\R^n$. This is an application of small ball estimate and Chernoff's method, that has been recently used in the context of Asymptotic Geometric Analysis in [1], [2].

math.FA↗
arXiv2007

An extension of a Bourgain--Lindenstrauss--Milman inequality

Let || . || be a norm on R^n. Averaging || (\eps_1 x_1, ..., \eps_n x_n) || over all the 2^n choices of \eps = (\eps_1, ..., \eps_n) in {-1, +1}^n, we obtain an expression ||| . ||| which is an unconditional norm on R^n. Bourgain, Lindenstrauss and Milman showed that, for a certain (large) constant η> 1, one may average over (ηn) (random) choices of \eps and obtain a norm that is isomorphic to ||| . |||. We show that this is the case for any η> 1.

math.FA↗
arXiv2006

Polynomial bounds for large Bernoulli sections of $\ell_1^N$

We prove a quantitative version of the bound on the smallest singular value of a Bernoulli covariance matrix (due to Bai and Yin). Then we use this bound, together with several recent developments, to show that the distance from a random (1-delta) n - dimensional section of ell_1^n, realised as an image of a sign matrix, to an Euclidean ball is polynomial in 1/delta (and independent of n), with high probability.

math.FA↗