Existence of similar point configurations in thin subsets of $${\mathbb {R}}^d$$
Existence of similar point configurations in thin subsets of $${\mathbb {R}}^d$$
复制标题
$${mathbb {R}}^d$$ 的薄子集中存在相似的点配置
DOI：
10.1007/s00209-020-02537-1
复制
发表时间：
2021
影响因子：
0.8
通讯作者：
Mkrtchyan, Sevak
中科院分区：
文献类型：
--
作者：
Greenleaf, Allan;Iosevich, Alex;Mkrtchyan, Sevak
我们证明了欧几里得空间中分数豪斯多夫维数集中存在相似和多重相似的点配置（或单纯形）。令 d ≥ 2 d≥ 2 且 E ⊂ R^ d E⊂ R d 为紧集。对于 k ≥ 1 k≥ 1，定义 Δ _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 ⊂ E\right\} ⊂ R^ k (k+ 1)/2, Δk(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 ≤ d k≤ d，这是 E 中 (k+ 1)(k+ 1) 点配置的同余集（根据排列）；对于 k> d k> d，它是顶点在 E 中的 (k+ 1)(k+ 1) 图的边长集合。许多作者之前的工作已经发现了值 s_ k, d< d sk, d< d，因此如果 E 的豪斯多夫维数满足\dim _\mathcal H (E)> s_ k, d dim H (E)> sk， d，则 Δ _k (E) Δ k (E) 具有正勒贝格测度。在本文中，我们研究了 Δ _k (E) Δ k (E) 的更精细的性质，即相似或多重相似构型的存在性。对于 r ε R,\, r> 0 rε R, r> 0，令 Δ _ k^ r (E):=\left {t\, ε Δ _k\left (E\right)： r t\, ε Δ _k\left (E\right)\right\} ⊂ Δ _k\left (E\right)。 Δ kr (E):= t∈ Δ k E: rtε 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 Δ _k (E) Δ k (E) 上的自然测度 ν _k ν k，则具有所有 r ε R _+ rε R+。因此，在 E 中存在许多对 (k+ 1)(k+ 1) 点配置，它们在比例因子 r 上相似。我们对此进行扩展以证明任何多重性的多重相似配置的存在。这些结果可以被视为对于紧薄集、R^d R d 中正密度集的 Furstenberg、Katznelson 和 Weiss 7、Bourgain 2 和 Ziegler 11 定理的变体和扩展。
We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff dimension in Euclidean space. Let d ≥ 2 d≥ 2 and E ⊂ R^ d E⊂ R d be a compact set. For k ≥ 1 k≥ 1, define Δ _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 ⊂ E\right\} ⊂ R^ k (k+ 1)/2, Δ k (E)=| x 1-x 2|,⋯,| xi-xj|,⋯,| xk-xk+ 1|: xii= 1 k+ 1⊂ E⊂ R k (k+ 1)/2, the (k+ 1)(k+ 1)-point configuration set of E. For k ≤ 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< d sk, d< d so that if the Hausdorff dimension of E satisfies\dim _\mathcal H (E)> s_ k, d dim H (E)> sk, d, then Δ _k (E) Δ k (E) has positive Lebesgue measure. In this paper we study more refined properties of Δ _k (E) Δ k (E), namely the existence of similar or multi–similar configurations. For r ∈ R,\, r> 0 r∈ R, r> 0, let Δ _ k^ r (E):=\left {t\, ∈ Δ _k\left (E\right): r t\, ∈ Δ _k\left (E\right)\right\} ⊂ Δ _k\left (E\right). Δ kr (E):= t∈ Δ k E: rt∈ Δ k E⊂ Δ k E. We show that if\dim _\mathcal H (E)> s_ k, d dim H (E)> sk, d, for a natural measure ν _k ν k on Δ _k (E) Δ k (E), one has all r ∈ 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 R^ d R d.