Existence of similar point configurations in thin subsets of $\Bbb R^d$

We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff measure in Euclidean space. These results can be viewed as variants, for thin sets, of theorems for sets of positive density in $\Bbb R^d$ due to Furstenberg, Katznelson and Weiss \cite{FKW90}, Bourgain \cite{B86} and Ziegler \cite{Z06}. Let $d \ge 2$ and $E\subset {\Bbb R}^d$ be a compact set. For $k\ge 1$, define $$\Delta_k(E)=\left\{\left(|x^1-x^2|, \dots, |x^i-x^j|,\dots, |x^k-x^{k+1}|\right): \left\{x^i\right\}_{i=1}^{k+1}\subset E\right\} \subset {\Bbb R}^{k(k+1)/2},$$ the {\it $(k+1)$-point configuration set} of $E$. For $k\le d$, this is (up to permutations) the set of congruences of $(k+1)$-point configurations in $E$; for $k>d$, it is the edge-length set of $(k+1)$-graphs whose vertices are in $E$. Previous works by a number of authors have found values $s_{k,d}<d$ so that if the Hausdorff dimension of $E$ is $>s_{k,d}$, then $\Delta_k(E)$ has positive Lebesgue measure. In this paper we study more refined properties of $\Delta_k(E)$, namely the existence of (exactly) similar or multi--similar configurations. For $r\in\Bbb R,\, r>0$, let $$\Delta_{k}^{r}(E):=\left\{\vec{t}\in \Delta_k\left(E\right): r\vec{t}\in \Delta_k\left(E\right)\right\}\subset \Delta_k\left(E\right).$$ We show that for all $E$ with Hausdorff dimension $>s_{k,d}$, a natural measure $\nu_k$ on $\Delta_k(E)$ and all $r\in\Bbb R_+$, one has $\nu_k\left(\Delta_{k}^{r}\left(E\right)\right)>0$. Thus, there exist many pairs, $\{x^1, x^2, \dots, x^{k+1}\}$ and $\{y^1, y^2, \dots, y^{k+1}\}$, in $E$ which are similar by the scaling factor $r$. We also show the existence of triply-similar and multi-similar configurations.