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

Abstract

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 $$Δ_k(E)=\left\{\left(|x^1-x^2|, \dots, |x^i-x^j|,\dots, |x^k-x^{k+1}|\right): \left\{x^i\right\}_{i=1}^{k+1}\subset E\right\} \subset {\Bbb R}^{k(k+1)/2}, $$ 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 $s_{k,d} s_{k,d}$, then $Δ_k(E)$ has positive Lebesgue measure. In this paper we study more refined properties of $Δ_k(E)$, namely the existence of (exactly) similar or multi--similar configurations. For $r\in\Bbb R,\, r>0$, let $$Δ_{k}^{r}(E):=\left\{\vec{t}\in Δ_k\left(E\right): r\vec{t}\in Δ_k\left(E\right)\right\}\subset Δ_k\left(E\right).$$ We show that for all $E$ with Hausdorff dimension $>s_{k,d}$, a natural measure $ν_k$ on $Δ_k(E)$ and all $r\in\Bbb R_+$, one has $ν_k\left(Δ_{k}^{r}\left(E\right)\right)>0$. 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.