Divisor-sum fibers

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) =\sum_{d\mid n,~d<n}d$.
Erdös--Granville--Pomerance--Spiro conjectured that, for any set $\mathcal{A}$ of asymptotic
density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak
form of this conjecture. In particular, we show that the EGPS conjecture holds for
infinite sets with counting function $O(x^{\frac12 + \epsilon(x)})$. We also disprove a hypothesis
from the same paper of EGPS by showing that for any positive numbers $\alpha$ and $\epsilon$,
there are integers $n$ with arbitrarily many $s$-preimages lying between $\alpha(1-\epsilon)n$
and $\alpha(1+\epsilon)n$. This talk is based on joint work with Paul Pollack and Carl Pomerance.