Realizing trees of configurations in thin sets

Allan Greenleaf, Alex Iosevich and Krystal Taylor

Vol. 335 (2025), No. 2, 355–372

DOI: 10.2140/pjm.2025.335.355

Abstract

Let $\varphi (x,y)$ be a continuous function, smooth away from the diagonal, such that, for some $\alpha >0$, the associated generalized Radon transforms

$$R_t^{\varphi }f(x)={\int \; <!--nolimits-->}_{\varphi (x,y)=t}f(y)\psi (y)d{\sigma }_{x,t}(y)$$

map $L^2({ℝ}^d)\to L_{\alpha }^2({ℝ}^d)$ for all $t>0$. Let $E$ be a compact subset of ${ℝ}^d$ for some $d\ge 2$, and suppose that the Hausdorff dimension of $E$ is greater than $d-\alpha$. We show that any tree graph $T$ on $k+1$ ($k\ge 1$) vertices is stably realizable in $E$, in the sense that for each $t$ in some open interval there exist distinct $x^1,x^2,\dots <!--FUNCTION\; APPLICATION-->,x^{k+1}\in E$ such that the $\varphi$-distance $\varphi (x^i,x^j)$ equals $t$ 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.

Keywords

finite point configurations, generalized Radon transforms, distance graph

Mathematical Subject Classification

Primary: 28A75, 42B35

Milestones

Received: 22 January 2024

Revised: 14 March 2025

Accepted: 18 March 2025

Published: 12 May 2025