Why Distance Two Is Exceptional for Poisson-Binomial Point Probabilities

Let $S$ be a finite sum of independent Bernoulli random variables. We ask how large the mean must be before the atom at $k+r$ can strictly exceed the atom at $k$. For the threshold $T_{k,r}:=\inf\{\mathbb{E}S:\mathbb{P}(S=k+r)>\mathbb{P}(S=k)\}$, the adjacent case recovers the classical Darroch boun