  We provide a counterexample to a conjecture of Hildebrand which states that
if $\mathcal{S}$ has positive lower density and for all $d$, $n$ is in
$\mathcal{S}$ if and only if $dn$ is in $\mathcal{S}$ for almost all $n$, then
$\mathcal{S} \cap (\mathcal{S}+1) \cap (\mathcal{S}+2)$ has positive lower
density and in particular is nonempty. Heuristically, we define $\mathcal{S}$
by $3^\ell n \in \mathcal{S}$ if and only if $\text{Re}(n^{iT} \chi(n)) > 0$
where $\chi$ is the nontrivial multiplicative character mod 3 and $T$ is chosen
such that $p^{iT} \approx \chi(p)$.
