Realizing trees of configurations in thin sets
Allan Greenleaf, Alex Iosevich, Krystal Taylor
TL;DR
The paper develops a general framework to realize finite point configurations, particularly trees, inside sets of prescribed Hausdorff dimension via generalized distance graphs and Sobolev bounds for generalized Radon transforms. Central to the approach is a graph-realization criterion (Theorem maingeneral) that reduces realizability to two operator bounds on a symmetric kernel derived from a configuration function, plus a positivity condition for an edge-configuration integral. The authors apply this to trees decorated with triangles, obtaining stable realizability thresholds such as ${ m dim}_{ m H}(E) > (2d+3)/3$ for congruent triangles in $d\ge 4$, and ${ m dim}_{ m H}(E) > 5/3$ for equi-area triangles in the plane, among other results. The framework unifies and extends prior work on Mattila–Sjölin configurations and provides a versatile method for translating harmonic-analytic bounds into combinatorial realizability results on fractal sets and manifolds.
Abstract
Let $φ(x,y)$ be a continuous function, smooth away from the diagonal, such that, for some $α>0$, 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 $L^2({\mathbb R}^d) \to L^2_α({\mathbb R}^d)$ 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 $>d-α$. 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 $φ$-distance $φ(x^i, x^j)=t$ 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.