Lutz和Stull两个定理的组合证明
Combinatorial proofs of two theorems of Lutz and Stull
AI总结：
本文用组合几何方法（离散位势理论、鸽巢原理、Katz-Tao引理）证明了Lutz-Stull的Marstrand型投影定理，并推广到任意m维平面投影。
摘要：
最近，Lutz和Stull使用算法信息论的方法证明了两个新的Marstrand型投影定理，这些定理涉及欧几里得空间中不假定为Borel甚至解析的子集。其中一个定理指出，如果$K \subset \mathbb{R}^{n}$是任何具有相等Hausdorff维数和填充维数的集合，那么对于几乎所有的$e \in S^{n - 1}$，有$$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$。这里$π_{e}$表示到$\mathrm{span}(e)$的正交投影。本文的主要目的是提供Lutz和Stull投影定理的证明，这些证明不涉及信息论概念，而是依赖于组合几何论证，例如Kaufman的“位势理论”方法的离散版本、鸽巢原理以及Katz和Tao的一个引理。次要目的是对Lutz和Stull的定理进行轻微推广：本文中的版本适用于$\mathbb{R}^{n}$中到$m$-平面的正交投影，对于所有$0 < m < n$。
英文摘要：
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman's "potential theoretic" method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to slightly generalise Lutz and Stull's theorems: the versions in this paper apply to orthogonal projections to $m$-planes in $\mathbb{R}^{n}$, for all $0 < m < n$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
