We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff dimension in Euclidean space. Let $$d \ge 2$$ d ≥ 2 and $$E\subset {{{\mathbb {R}}} }^d$$ E ⊂ R d be a compact set. For $$k\ge 1$$ k ≥ 1 , define $$\begin{aligned} \Delta _k(E)=\left\{ \left( |x^1-x^2|, \dots , |x^i-x^j|,\dots , |x^k-x^{k+1}|\right) : \left\{ x^i\right\} _{i=1}^{k+1}\subset E\right\} \subset {{{\mathbb {R}}} }^{k(k+1)/2}, \end{aligned}$$ Δ k ( E ) = | x 1 - x 2 | , ⋯ , | x i - x j | , ⋯ , | x k - x k + 1 | : x i i = 1 k + 1 ⊂ E ⊂ R k ( k + 1 ) / 2 , the $$(k+1)$$ ( k + 1 ) -point configuration set of E . For $$k\le d$$ k ≤ d , this is (up to permutations) the set of congruences of $$(k+1)$$ ( k + 1 ) -point configurations in E ; for $$k>d$$ k > d , it is the edge-length set of $$(k+1)$$ ( k + 1 ) -graphs whose vertices are in E . Previous works by a number of authors have found values $$s_{k,d}s_{k,d}$$ dim H ( E ) > s k , d , then $$\Delta _k(E)$$ Δ k ( E ) has positive Lebesgue measure. In this paper we study more refined properties of $$\Delta _k(E)$$ Δ k ( E ) , namely the existence of similar or multi–similar configurations. For $$r\in {\mathbb {R}},\, r>0$$ r ∈ R , r > 0 , let $$\begin{aligned} \Delta _{k}^{r}(E):=\left\{ \mathbf {t}\,\in \Delta _k\left( E\right) : r\mathbf {t}\,\in \Delta _k\left( E\right) \right\} \subset \Delta _k\left( E\right) . \end{aligned}$$ Δ k r ( E ) : = t ∈ Δ k E : r t ∈ Δ k E ⊂ Δ k E . We show that if $$\dim _{\mathcal H}(E)>s_{k,d}$$ dim H ( E ) > s k , d , for a natural measure $$\nu _k$$ ν k on $$\Delta _k(E)$$ Δ k ( E ) , one has all $$r\in {\mathbb {R}}_+$$ r ∈ R + . Thus, in E there exist many pairs of $$(k+1)$$ ( 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 $${\mathbb {R}}^d$$ R d .

中文翻译：

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$${\mathbb {R}}^d$$ 的细子集中相似点配置的存在

我们证明了欧几里得空间中分数豪斯多夫维数集合中相似和多相似点配置（或单纯形）的存在。令 $$d \ge 2$$ d ≥ 2 和 $$E\subset {{{\mathbb {R}}} }^d$$ E ⊂ R d 是一个紧集。对于 $$k\ge 1$$ k ≥ 1 ，定义 $$\begin{aligned} \Delta _k(E)=\left\{ \left( |x^1-x^2|, \dots , |x ^ix^j|,\dots , |x^kx^{k+1}|\right) : \left\{ x^i\right\} _{i=1}^{k+1}\subset E \right\} \subset {{{\mathbb {R}}} }^{k(k+1)/2}, \end{aligned}$$ Δ k ( E ) = | x 1 - x 2 | , ⋯ , | xi - xj | , ⋯ , | xk - xk + 1 | : xii = 1 k + 1 ⊂ E ⊂ R k ( k + 1 ) / 2 ，E 的 $$(k+1)$$ ( k + 1 ) -点配置集。对于 $$k\le d$$ k ≤ d ，这是（最多排列）E 中 $$(k+1)$$ ( k + 1 ) -点配置的同余集；对于 $$k>d$$ k > d ，它是顶点在 E 中的 $$(k+1)$$ ( k + 1 ) -图的边长集。许多作者以前的作品已经发现值 $$s_{k,d}s_{k,d}$$ dim H ( E ) > sk , d ，然后 $$\Delta _k(E)$$ Δ k ( E ) 具有正 Lebesgue 测度。在本文中，我们研究了 $$\Delta _k(E)$$ Δ k ( E ) 的更精细的属性，即相似或多相似配置的存在。对于 $$r\in {\mathbb {R}},\, r>0$$ r ∈ R , r > 0 ，令 $$\begin{aligned} \Delta _{k}^{r}(E) :=\left\{ \mathbf {t}\,\in \Delta _k\left( E\right) : r\mathbf {t}\,\in \Delta _k\left( E\right) \right\} \subset \Delta _k\left( E\right) 。\end{aligned}$$ Δ kr ( E ) : = t ∈ Δ k E : rt ∈ Δ k E ⊂ Δ k E 。我们证明如果 $$\dim _{\mathcal H}(E)>s_{k,d}$$ dim H ( E ) > sk , d ，对于自然测度 $$\nu _k$$ ν k on $$\Delta _k(E)$$ Δ k ( E ) , 一个有所有 $$r\in {\mathbb {R}}_+$$ r ∈ R + 。因此，在 E 中存在许多对 $$(k+1)$$ ( k + 1 ) -点配置，它们与比例因子 r 相似。我们将其扩展以显示任何多重性的多重相似配置的存在。这些结果可以看作是紧致薄集的变体和扩展，Furstenberg、Katznelson 和 Weiss [7]、Bourgain [2] 和 Ziegler [11] 的定理在 $${\mathbb {R} 中的正密度集合}^d$$ R d 。