# Configuration sets with nonempty interior

## TL;DR

This paper generalizes classical geometric measure theory results, demonstrating that certain configuration sets derived from sets with sufficiently large Hausdorff dimension have nonempty interior, under regularity conditions on associated Radon transforms.

## Contribution

It introduces a unified framework for analyzing the interior of configuration sets using generalized Radon transforms and extends classical results to more general $\

## Key findings

- Configuration sets with large Hausdorff dimension have nonempty interior.
- Generalized Radon transform regularity implies interiority of configuration sets.
- Results extend to configurations involving two different sets.

## Abstract

A theorem of Steinhaus states that if $E\subset \mathbb R^d$ has positive Lebesgue measure, then the difference set $E-E$ contains a neighborhood of $0$. Similarly, if $E$ merely has Hausdorff dimension $\dim_{\mathcal H}(E)>(d+1)/2$, a result of Mattila and Sj\"olin states that the distance set $\Delta(E)\subset\mathbb R$ contains an open interval. In this work, we study such results from a general viewpoint, replacing $E-E$ or $\Delta(E)$ with more general $\Phi\,$-configurations for a class of $\Phi:\mathbb R^d\times\mathbb R^d\to\mathbb R^k$, and showing that, under suitable lower bounds on $\dim_{\mathcal H}(E)$ and a regularity assumption on the family of generalized Radon transforms associated with $\Phi$, it follows that the set $\Delta_\Phi(E)$ of $\Phi$-configurations in $E$ has nonempty interior in $\mathbb R^k$. Further extensions hold for $\Phi\,$-configurations generated by two sets, $E$ and $F$, in spaces of possibly different dimensions and with suitable lower bounds on $\dim_{\mathcal H}(E)+\dim_{\mathcal H}(F)$.