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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.
The classical ErdH{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,dots,a_nin mathbb{R}^d$, any $xin mathbb{R}^d$, and uniformly random $(xi_1,dots,xi_n)in{-1,1}^n$, we have $Pr(a_1xi_1+dots+a_nxi_n=x)=O(n^{-1/2})$. In this paper we
We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and asymptotic to $frac{8sqrt{3}}{
This paper is devoted to a new family of reverse Hardy-Littlewood-Sobolev inequalities which involve a power law kernel with positive exponent. We investigate the range of the admissible parameters and characterize the optimal functions. A striking o
We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in
The Newell-Littlewood numbers $N_{mu, u,lambda}$ are tensor product multiplicities of Weyl modules for classical Lie groups, in the stable limit. For which triples of partitions $(mu, u,lambda)$ does $N_{mu, u,lambda}>0$ hold? The Littlewood-Richards