  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 $\dim_{\mathcal H}(E)+\dim_{\mathcal H}(F)$.
