Realizing trees of configurations in thin sets

Abstract

Let $\ddot{I}•(x,y)$ be a continuous function, smooth away from the diagonal, such that, for some $\hat{I}\pm >0$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^ϕf(x)=\int_{ϕ(x,y)=t} f(y) ψ(y) dσ_{x,t}(y) \end{equation} map $L^2({\mathbb{R}}^d)\to L_{\hat{I}}^2\pm ({\mathbb{R}}^d)$ 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 $>d-\hat{I}\pm$. 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 $\ddot{I}•$-distance $\ddot{I}•(x^i,x^j)=t$ 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.