Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A sharp inverse Littlewood-Offord theorem   Let $\eta_i, i=1,..., n$ be iid Bernoulli random variables. Given a multiset
$\bv$ of $n$ numbers $v_1, ..., v_n$, the \emph{concentration probability}
$\P_1(\bv)$ of $\bv$ is defined as $\P_1(\bv) := \sup_{x} \P(v_1 \eta_1+ ...
v_n \eta_n=x)$. A classical result of Littlewood-Offord and Erd\H os from the
1940s asserts that if the $v_i $ are non-zero, then this probability is at most
$O(n^{-1/2})$. Since then, many researchers obtained better bounds by assuming
various restrictions on $\bv$.
  In this paper, we give an asymptotically optimal characterization for all
multisets $\bv$ having large concentration probability. This allow us to
strengthen or recover several previous results in a straightforward manner.

35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con