Multilinear generalized Radon transforms and point configurations

TL;DR

This paper develops a multilinear framework for Radon transforms to analyze geometric configurations in fractal sets, establishing conditions under which these configurations have positive measure, and applies this to problems in geometric measure theory and discrete geometry.

Contribution

It introduces a graph-theoretic approach to multilinear Radon transforms, extending results on point configurations and Falconer-type problems to a broader, more general setting.

Findings

Positive Lebesgue measure for configuration sets when Hausdorff dimension exceeds a threshold

Extension of Falconer conjecture results to multilinear and higher-point configurations

Applications to Erdős-type problems and fractal regular value theorems

Abstract

We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving $(k+1)$-point configurations in geometric measure theory, with $k \ge 2$, including the distribution of simplices, volumes and angles determined by the points of fractal subsets $E \subset {\Bbb R}^d$, $d \ge 2$. If $T_k(E)$ denotes the set of noncongruent $(k+1)$-point configurations determined by $E$, we show that if the Hausdorff dimension of $E$ is greater than $d-\frac{d-1}{2k}$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. This compliments previous work on the Falconer conjecture (\cite{Erd05} and the references there), as well as work on finite point configurations \cite{EHI11,GI10}. We also give applications to Erd\"os-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in \cite{EIT11}.