VC-Dimension and Distance Chains in $\mathbb{F}_q^d$
Ruben Ascoli, Livia Betti, Justin Cheigh, Alex Iosevich, Ryan Jeong, Xuyan Liu, Brian McDonald, Wyatt Milgrim, Steven J. Miller, Francisco Romero Acosta, Santia… 展开作者
Ruben Ascoli, Livia Betti, Justin Cheigh, Alex Iosevich, Ryan Jeong, Xuyan Liu, Brian McDonald, Wyatt Milgrim, Steven J. Miller, Francisco Romero Acosta, Santiago Velazquez Iannuzzelli
Given a domain $X$ and a collection $\mathcal{H}$ of functions $h:X\to \{0,1\}$, the Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions $\mathcal{H}_t^{'2}(E): \mathbb{F}_q^2\to \{0,1\}$, corresponding to indicator functions of circles centered at points in a subset $E\subseteq \mathbb{F}_q^2$. They showed that when $|E|$ is large enough, the VC-dimension of $\mathcal{H}_t^{'2}(E)$ is the same as in the case that $E = \mathbb F_q^2$. We study a related hypothesis class, $\mathcal{H}_t^d(E)$, corresponding to intersections of spheres in $\mathbb{F}_q^d$, and ask how large $E\subseteq \mathbb{F}_q^d$ needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever $|E|\geq C_dq^{d-1/(d-1)}$ for $d\geq 3$, the VC-dimension of $\mathcal{H}_t^d(E)$ is as large as possible. We get a slightly stronger result if $d=3$: this result holds as long as $|E|\geq C_3 q^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq C_2 q^{7/4}$.
Improvement on $2$-chains inside thin subsets of Euclidean spaces
证明了若$E\subset\mathbb{R}^d$，$d\geq 2$的Hausdorff维数大于$\frac{d}{2}+\frac{1}{3}$，则$E$，$$Δ_2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset\mathbb{R}^2$$中$2$ -链的间隙集具有正的Lebesgue测度.推广了Wolff-Erdogan关于距离的结果，改进了Bennett，Iosevich和Taylor关于有限链的一个结果.我们还考虑了$2$ -链的相似类，$$S_2(E)=\left\{\frac{t_1}{t_2}:(t_1,t_2)\inΔ_2(E)\right\}=\left\{\frac{|x-y|}{|y-z|}: x, y, z\in E \right\}\subset\mathbb{R},$$，并证明了当$|S_2(E)|>0$.
