TITLE: Anti-concentration of Bernoulli sums
QUESTION [19 upvotes]: Let $a_1,\ldots,a_n$ be real numbers such that $\sum_i a_i^2 =1$ and let $X_1,\ldots,X_n$ be independent, uniformly distributed, Bernoulli $\pm 1$ random variables. Define the random variable
$S:= \sum_i X_i a_i $
Are there absolute constants $\epsilon >0$ and $\delta<1$ such that for every $a_1,\ldots,a_n$
${\mathbb P} [|S| \leq \epsilon ] \leq \delta$ ?

REPLY [3 votes]: Your question is part of what's called Littlewood-Offord theory, which has seen a lot of progress lately in work of Tao and Vu and of Rudelson and Vershynin.  Take a look at Section 1.2 and especially Theorem 1.5 of this paper by Rudelson and Vershynin for more precise results than in George's answer.  (Incidentally, that paper also contains arguments based on the central limit theorem, in the Berry-Esseen form, along the lines of what Igor suggests.)