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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

$S:= \sum_i X_i a_i$

Is it true (with no further assumption on $a_1,\ldots,a_n$) that $S$ is unlikely to be small and that, specifically,

Are there are positive absolute constants $c_1, c_2, c_3$ \epsilon >0$and$\delta<1$such that for every$a_1,\ldots,a_n{\mathbb P} [|S| \leq \epsilon ] \leq c_1\cdot \epsilon + c_2 \cdot n^{-c_3}$delta$ ?

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# Anti-concentration of bernoulli sums

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

$S:= \sum_i X_i a_i$

Is it true (with no further assumption on $a_1,\ldots,a_n$) that $S$ is unlikely to be small and that, specifically, there are positive constants $c_1, c_2, c_3$ such that

${\mathbb P} [|S| \leq \epsilon ] \leq c_1\cdot \epsilon + c_2 \cdot n^{-c_3}$ ?