The point-to-set principle \cite{LutLut17} characterizes the Hausdorff dimension of a subset $E\subseteq\R^n$ by the \textit{effective} (or algorithmic) dimension of its individual points. This characterization has been used to prove several results in classical, i.e., without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand's projection theorem, a fundamental result in fractal geometry. In this paper, we introduce an extension of point-to-set principle - the notion of \textit{optimal oracles} for subsets $E\subseteq\R^n$. One of the primary motivations of this definition is that, if $E$ has optimal oracles, then the conclusion of Marstrand's projection theorem holds for $E$. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of $E$ agree, then $E$ has optimal oracles. Moreover, we show that the existence of sufficiently nice outer measures on $E$ implies the existence of optimal Hausdorff oracles. In particular, the existence of exact gauge functions for a set $E$ is sufficient for the existence of optimal Hausdorff oracles, and is therefore sufficient for Marstrand's theorem. Thus, the existence of optimal oracles extends the currently known sufficient conditions for Marstrand's theorem to hold. Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a generalization of Davies theorem on projections.
翻译:点到确定的原则 \ cite {LutLut17} 代表着一个子集 $E\ subseteq\ R} 的Hausdorf 维度的特性。 这个特性被用来在古典中证明若干结果, 也就是说, 在没有任何可计算性要求的情况下, 分析。 最近的工作显示, 算法技术可以有成效地适用于 Marstrand 的投影符, 这是最佳几何测量的一个根本结果。 在本文中, 我们引入了一个点到定原则的延伸, 由\ textit@ optime} (或算法性) 维度的扩展 。 这个定义的主要动机之一是, 如果$是最佳的, 那么, Marstrand 的算法将标定值用于一般的 $。 我们显示, 每一个解析集都有最佳的 。 如果Haustorff 和 $ 确定原则的维度是最佳的, 则由目前最优的 或最优的 美元 。