论文标题
卢茨和Stull的两个定理的组合证明
Combinatorial proofs of two theorems of Lutz and Stull
论文作者
论文摘要
最近，卢茨和Stull使用算法信息理论的方法证明了两个新的Marstrand型投影定理，涉及欧几里得空间的子集，这些子集并非被认为是Borel，甚至是分析性的。一个定理之一指出，如果$ k \ subset \ mathbb {r}^{n} $是任何具有均等的hausdorff和包装尺寸的设置，则$ \ dim _ {\ dim _ {\ mathrm {h}}π_{e}（e}（e}（k）（k）= \ \ \ \ \ \ { $$对于S^{n -1} $中的几乎每个$ e \。这里$π_{e} $代表正交投影到$ \ mathrm {span}（e）$。 本文的主要目的是为Lutz和Stull的投影定理提供证明，这些证明并不指信息理论概念。取而代之的是，他们将依靠组合几何论证，例如Kaufman的“潜在理论”方法的离散版本，Pigonhole原理以及Katz和Tao的引理。次要目的是稍微概括lutz和Stull的定理：本文中的版本适用于$ 0 <m <n $的$ \ mathbb {r}^{n} $中的$ m $ planes。
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \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 $0 < m < n$.
