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Claim. P. Pollack, C. Pomerance and L. Thompson, Divisor-sum fibers, Mathematika 64 (2018), 330--342, Theorem 1.2: "Let $\epsilon=\epsilon(x)$ be a fixed function tending to 0 as $x\to\infty$. Suppose that $\mathcal{A}$ is a set of at most $x^{1/2+\epsilon(x)}$ positive integers. Then, as $x\to\infty$, $\#\{n\leq x:s(n)\in\mathcal{A}\}=o_\epsilon(x)$, uniformly in the choice of $\mathcal{A}$" (result page).

Derived here, not stated as a theorem in the paper: if a fixed set $B$ has $|B\cap[1,y]|\le y^{1/2+o(1)}$, then since $s(n)<2n\log\log n$ for large $n$, applying the theorem to $B\cap[1,2x\log\log x]$ shows that $s^{-1}(B)$ has density zero, the assertion of Problem 955 for $B$. The paper uses the same truncation to recover the palindrome theorem.

Covers. Every $A$ with $|A\cap[1,y]|\le y^{1/2+o(1)}$; the general assertion stays open.

Depends on. No page of this wiki.

Acceptance. Refereed: Mathematika. The site's commentary credits the result, but the site labels the problem OPEN, so no curator acceptance is listed.