Let $w\ll n$ (say $w=n^{0.1}$) and $a\_1,\ldots,a\_w$ be positive real numbers such that $\sum\_{i \in w} a\_i=n$. Also, let $x\_1,x\_2,\ldots, x\_w$ be i.i.d. $\pm 1$ random variables. What is the best $t$ such that one always has 
$$
\Pr\_{x\_1,\ldots, x\_w} [|\sum\_{i\le w} a\_i x\_i| \ge t] \ge 0.01
$$



Anti-concentration about the mean for sum of Bernoulli random variables
The best $t$ I do not know. But one can for example get some bounds by applying the Paley-Zygmund inequality to the random variable $Z=(\sum\_i a\_i X\_i)^2$.


