A Sharp Estimate for Divisors of Bernoulli Sums

TL;DR

This paper provides a precise estimate for the probability that a sum of Bernoulli random variables is divisible by a given integer, with bounds that improve understanding of divisibility properties in probabilistic sums.

Contribution

The paper introduces a sharp asymptotic estimate for the divisibility probabilities of Bernoulli sums, refining previous bounds with explicit error terms.

Findings

The supremum of the difference between the divisibility probability and its approximation is bounded by O(log^{5/2} n / n^{3/2})

The approximation E(n,d) is tightly bounded by functions involving the residue of n modulo 2d

The results improve understanding of the distribution of Bernoulli sums modulo integers.

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 $$\sup_{2\le d\le n} \big|\P\big\{d|S_n\big\}- E(n,d) \big|= {\cal O}\big({\log^{5/2} n \over n^{3/2}}\big), $$ 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.