欧几里得空间薄子集上相似点构型的存在性
经典分析与常微分方程
2021-04-28 v2
摘要
我们证明了欧几里得空间中分数阶 Hausdorff 测度集合内相似与多重相似点构型（或单形）的存在性。这些结果可视为 Furstenberg、Katznelson 和 Weiss \cite{FKW90}、Bourgain \cite{B86} 与 Ziegler \cite{Z06} 关于 R d \Bbb R^d R d d ≥ 2 d \ge 2 d ≥ 2 E ⊂ R d E\subset {\Bbb R}^d E ⊂ R d k ≥ 1 k\ge 1 k ≥ 1 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 , \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}, 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 , E E E ( k + 1 ) (k+1) ( k + 1 ) k ≤ d k\le d k ≤ d E E E ( k + 1 ) (k+1) ( k + 1 ) k > d k>d k > d E E E ( k + 1 ) (k+1) ( k + 1 ) s k , d < d s_{k,d}<d s k , d < d E E E > s k , d >s_{k,d} > s k , d Δ k ( E ) \Delta_k(E) Δ k ( E ) Δ k ( E ) \Delta_k(E) Δ k ( E ) r ∈ R , r > 0 r\in\Bbb R,\, r>0 r ∈ R , r > 0 Δ k r ( E ) : = { t ⃗ ∈ Δ k ( E ) : r t ⃗ ∈ Δ k ( E ) } ⊂ Δ k ( E ) . \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). Δ k r ( E ) := { t ∈ Δ k ( E ) : r t ∈ Δ k ( E ) } ⊂ Δ k ( E ) . > s k , d >s_{k,d} > s k , d E E E Δ k ( E ) \Delta_k(E) Δ k ( E ) ν k \nu_k ν k r ∈ R + r\in\Bbb R_+ r ∈ R + ν k ( Δ k r ( E ) ) > 0 \nu_k\left(\Delta_{k}^{r}\left(E\right)\right)>0 ν k ( Δ k r ( E ) ) > 0 E E E r r r { x 1 , x 2 , … , x k + 1 } \{x^1, x^2, \dots, x^{k+1}\} { x 1 , x 2 , … , x k + 1 } { y 1 , y 2 , … , y k + 1 } \{y^1, y^2, \dots, y^{k+1}\} { y 1 , y 2 , … , y k + 1 }
引用
@article{arxiv.1808.04290,
title = {Existence of similar point configurations in thin subsets of $\Bbb R^d$},
author = {Allan Greenleaf and Alex Iosevich and Sevak Mkrtchyan},
journal= {arXiv preprint arXiv:1808.04290},
year = {2021}
}
备注
14 pages; end of Sec. 4 clarified; additional comment and reference added in Sec. 5