Abstract
We prove the following conjecture, due to Tomaszewski (1986): Let $X=
\sum_i=1^n a_i x_i$, where $\sum_i a_i^2=1$ and each $x_i$ is a
uniformly random sign. Then $\Pr|X|1 1/2$. Our main novel tools are
local concentration inequalities and an improved Berry-Esseen inequality for
Rademacher sums.
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