A Sharp Estimate for Divisors of Bernoulli Sums

Abstract

Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli r.v.'s. Let $0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, ${\bar{r}}_d(n)=2d-r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt n})[e^{- {r_d(n)^2/2 n}} +e^{- {\bar r_d(n)^2/2 n}}]$. We show that $$\underset{2\le d\le n}{\sup }|\P \{d|S_n\}-E(n,d)|=\mathcal{O}(\frac{{\log }^{5/2}n}{n^{3/2}}),$$ where $E(n,d)$ verifies $c_1\b(n,d)\le E(n,d)\le c_2\b(n,d) $ and $c_1,c_2$ are numerical constants.