Search for: All records

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 $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ 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 realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$. 

We prove new results of Mattila–Sjölin type, giving lower bounds on Hausdorff dimensions of thin sets E ⊂ R^d ensuring that various k-point configuration sets, generated by elements of E , have nonempty interior. The dimensional thresholds in our previous work (Greenleaf et al., Mathematika 68(1):163–190, 2022) were dictated by associating to a configuration function a family of generalized Radon transforms, and then optimizing L^2-Sobolev estimates for them over all nontrivial bipartite partitions of the k points. In the current work, we extend this by allowing the optimization to be done locally over the configuration’s incidence relation, or even microlocally over the conormal bundle of the incidence relation. We use this approach to prove Mattila–Sjölin type results for (i) areas of subtriangles determined by quadrilaterals and pentagons in a set E ⊂ R^2; (ii) pairs of ratios of distances of 4-tuples in R^d; and (iii) similarity classes of triangles in R^d, as well as to (iv) give a short proof of Palsson and Romero Acosta’s result on congruence classes of triangles in R .