Configuration sets with nonempty interior

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 Sjölin states that the distance set $Δ(E)\subset\mathbb R$ contains an open interval. In this work, we study such results from a general viewpoint, replacing $E-E$ or $Δ(E)$ with more general $Φ\,$-configurations for a class of $Φ:\mathbb R^d\times\mathbb R^d\to\mathbb R^k$, 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 $Φ$, it follows that the set $Δ_Φ(E)$ of $Φ$-configurations in $E$ has nonempty interior in $\mathbb R^k$. Further extensions hold for $Φ\,$-configurations generated by two sets, $E$ and $F$, in spaces of possibly different dimensions and with suitable lower bounds on $\dim_{\mathcal H}(E)+\dim_{\mathcal H}(F)$.