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The point-to-set principle \cite{LutLut17} characterizes the Hausdorffdimension of a subset $E\subseteq\R^n$ by the \textit{effective} (oralgorithmic) dimension of its individual points. This characterization has beenused to prove several results in classical, i.e., without any computabilityrequirements, analysis. Recent work has shown that algorithmic techniques canbe fruitfully applied to Marstrand's projection theorem, a fundamental resultin fractal geometry.  In this paper, we introduce an extension of point-to-set principle - thenotion of \textit{optimal oracles} for subsets $E\subseteq\R^n$. One of theprimary motivations of this definition is that, if $E$ has optimal oracles,then the conclusion of Marstrand's projection theorem holds for $E$. We showthat every analytic set has optimal oracles. We also prove that if theHausdorff 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, theexistence of exact gauge functions for a set $E$ is sufficient for theexistence of optimal Hausdorff oracles, and is therefore sufficient forMarstrand's theorem. Thus, the existence of optimal oracles extends thecurrently known sufficient conditions for Marstrand's theorem to hold.  Under certain assumptions, every set has optimal oracles. However, assumingthe axiom of choice and the continuum hypothesis, we construct sets which donot have optimal oracles. This construction naturally leads to a generalizationof Davies theorem on projections.

Optimal Oracles for Point-to-Set Principles