Let
$s(\cdot )$
denote the sum-of-proper-divisors function, that is,
$s(n)=\sum _{d\mid n,~d<n}d$
. Erdős, Granville, Pomerance, and Spiro conjectured that for any set
$\mathscr{A}$
of asymptotic density zero, the preimage set
$s^{-1}(\mathscr{A})$
also has density zero. We prove a weak form of this conjecture: if
$\unicode[STIX]{x1D716}(x)$
is any function tending to
$0$
as
$x\rightarrow \infty$
, and
$\mathscr{A}$
is a set of integers of cardinality at most
$x^{1/2+\unicode[STIX]{x1D716}(x)}$
, then the number of integers
$n\leqslant x$
with
$s(n)\in \mathscr{A}$
is
$o(x)$
, as
$x\rightarrow \infty$
. In particular, the EGPS conjecture holds for infinite sets with counting function
$O(x^{1/2+\unicode[STIX]{x1D716}(x)})$
. We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers
$\unicode[STIX]{x1D6FC}$
and
$\unicode[STIX]{x1D716}$
, there are integers
$n$
with arbitrarily many
$s$
-preimages lying between
$\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D716})n$
and
$\unicode[STIX]{x1D6FC}(1+\unicode[STIX]{x1D716})n$
. Finally, we make some remarks on solutions
$n$
to congruences of the form
$\unicode[STIX]{x1D70E}(n)\equiv a~(\text{mod}~n)$
, 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
$n\leqslant x$
, making it uniform in
$a$
.