  In this paper we prove that the set of tuples of edge lengths in $K_1\times
K_2$ corresponding to a finite tree has non-empty interior, where
$K_1,K_2\subset \mathbb{R}$ are Cantor sets of thickness $\tau(K_1)\cdot
\tau(K_2) >1$. Our method relies on establishing that the pinned distance set
is robust to small perturbations of the pin. In the process, we prove a
nonlinear version of the classic Newhouse gap lemma, and show that if $K_1,K_2$
are as above and $\phi: \mathbb{R}^2\times \mathbb{R}^2 \rightarrow \mathbb{R}
$ is a function satisfying some mild assumptions on its derivatives, then there
exists an open set $S$ so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has
non-empty interior.

Classical Analysis and ODEs