论文标题
扩展点对点原理的范围
Extending the Reach of the Point-to-Set Principle
论文作者
论文摘要
J. Lutz和N. Lutz（2018）的重点原理最近使计算理论可用于回答有关欧几里得空间中分形几何的开放问题，$ \ Mathbb {r}^n $。这些是经典的问题，这意味着它们的陈述不涉及逻辑的计算或相关方面。 在本文中，我们将点对点原理的范围从欧几里得空间扩展到任意可分离的度量空间$ x $。我们首先延长了两个分形维度 - 经典Hausdorff和包装尺寸的分形维度，分配了尺寸$ \ dim（x）$和$ \ textrm {dim}（x）$ to单个点$ x \ in x $中的单个点$ x \ - 向任意分离的分离空间和任意分离的计量家庭。然后，我们的前两个主要结果将点对点的原理扩展到了任意可分离的度量空间和大量的仪表家庭。 我们通过使用它来证明有关超级空间中经典分形维的新定理来证明我们扩展点对集合原理的力量。 （对于具体的计算示例，$ e_0，e_1，e_2，e_2，\ ldots $用于构造飞机上的自相似分形$ e $是飞机超空间的要素，它们在超级空间中融合到$ e $。紧凑型集的所有超空间上的经典上minkowski尺寸。我们使用此定理给出，对于分析性的所有集合$ e $，即$ \mathbfς^1_1 $，在$ e $的包装维度方面，$ e $的超空间的包装维度紧密。
The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that their statements do not involve computation or related aspects of logic. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces $X$. We first extend two fractal dimensions--computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions $\dim(x)$ and $\textrm{Dim}(x)$ to individual points $x\in X$--to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. (For a concrete computational example, the stages $E_0, E_1, E_2, \ldots$ used to construct a self-similar fractal $E$ in the plane are elements of the hyperspace of the plane, and they converge to $E$ in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We use this theorem to give, for all sets $E$ that are analytic, i.e., $\mathbfΣ^1_1$, a tight bound on the packing dimension of the hyperspace of $E$ in terms of the packing dimension of $E$ itself.
