Existence of similar point configurations in thin subsets of $${\mathbb {R}}^d$$

Abstract

We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff dimension in Euclidean space. Let \(d \ge 2\) and \(E\subset {{{\mathbb {R}}} }^d\) be a compact set. For \(k\ge 1\), define $$\begin{aligned} \Delta _k(E)=\left\{ \left( |x^1-x^2|, \dots , |x^i-x^j|,\dots , |x^k-x^{k+1}|\right) : \left\{ x^i\right\} _{i=1}^{k+1}\subset E\right\} \subset {{{\mathbb {R}}} }^{k(k+1)/2}, \end{aligned}$$the \((k+1)\)-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 satisfies \(\dim _{\mathcal H}(E)>s_{k,d}\), then \(\Delta _k(E)\) has positive Lebesgue measure. In this paper we study more refined properties of \(\Delta _k(E)\), namely the existence of similar or multi–similar configurations. For \(r\in {\mathbb {R}},\, r>0\), let $$\begin{aligned} \Delta _{k}^{r}(E):=\left\{ \mathbf {t}\,\in \Delta _k\left( E\right) : r\mathbf {t}\,\in \Delta _k\left( E\right) \right\} \subset \Delta _k\left( E\right) . \end{aligned}$$We show that if \(\dim _{\mathcal H}(E)>s_{k,d}\), for a natural measure \(\nu _k\) on \(\Delta _k(E)\), one has all \(r\in {\mathbb {R}}_+\). Thus, in E there exist many pairs of \((k+1)\)-point configurations which are similar by the scaling factor r. We extend this to show the existence of multi–similar configurations of any multiplicity. These results can be viewed as variants and extensions, for compact thin sets, of theorems of Furstenberg, Katznelson and Weiss [7], Bourgain [2] and Ziegler [11] for sets of positive density in \({\mathbb {R}}^d\).