97 research outputs found
Non-abelian Littlewood-Offord inequalities
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
Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices
Let denote a random symmetric by 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 is non-singular with probability for any positive
constant . 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
Geometric and o-minimal Littlewood-Offord problems
The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero
vectors , any , and uniformly
random , we have
. In this paper we show that
whenever is definable
with respect to an o-minimal structure (for example, this holds when is any
algebraic hypersurface), under the necessary condition that it does not contain
a line segment. We also obtain an inverse theorem in this setting
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