Realizing trees of configurations in thin sets

Abstract: Let $\phi(x,y)$ be a continuous function, smooth away from the diagonal, such that, for some $\alpha>0$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $L^2({\mathbb R}^d) \to L^2_{\alpha}({\mathbb R}^d)$ for all $t>0$. Let $E$ be a compact subset of ${\mathbb R}^d$ for some $d \ge 2$, and suppose that the Hausdorff dimension of $E$ is $>d-\alpha$. We show that any tree graph $T$ on $k+1$ ($k \ge 1$) vertices is \new{stably} realizable in $E$, in the sense that \new{for each $t$ in some open interval} there exist distinct $x^1, x^2, \dots, x^{k+1} \in E$ %and $t>0$ such that the $\phi$-distance $\phi(x^i, x^j)=t$ for all pairs $(i,j)$ corresponding to the edges of $T$. We extend this result to trees whose edges are prescribed by more complicated point configurations, such as congruence classes of triangles.