A counterexample to Hildebrand's conjecture on stable sets

We provide a counterexample to a conjecture of Hildebrand which states that if $\S$ has positive lower density and is stable i.e. for all $d$, $n$ is in $\mathcal{S}$ if and only if $dn$ is in $\mathcal{S}$ except on a set of density $0$ then $\mathcal{S} \cap (\mathcal{S}+1) \cap (\mathcal{S}+2)$ has positive lower density and in particular is nonempty. We further show there exists a stable set of density $1 -\frac{1}{q-1}$ such that $\mathcal{S} \cap \cdots \cap (\mathcal{S} + q -1) = \emptyset$ when $q$ is a prime, matching a bound proven by Hildebrand. Finally, we construct a function $f : \mathbb{N} \rightarrow \{\pm 1\}$ such that $f(pn) = -f(n)$ for all but a $0$ density set of $n$ depending on the prime $p$ but which fails the analogues of Sarnak and Chowla's conjectures.