Divisor-sum fibers

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erd\H{o}s-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: If $\epsilon(x)$ is any function tending to $0$ as $x\to\infty$, and $\mathcal{A}$ is a set of integers of cardinality at most $x^{\frac12+\epsilon(x)}$, then the number of integers $n\le x$ with $s(n) \in \mathcal{A}$ is $o(x)$, as $x\to\infty$. In particular, 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 $\alpha(1+\epsilon)n$. Finally, we make some remarks on solutions $n$ to congruences of the form $\sigma(n) \equiv a\pmod{n}$, proposing a modification of a conjecture appearing in recent work of the first two authors. We also improve a previous upper bound for the number of solutions $n \leq x$, making it uniform in $a$.