Projections of sets with optimal oracles onto $k$-planes

TL;DR

This paper establishes new projection estimates for sets with optimal oracles in Euclidean space, expanding the known conditions for Marstrand's theorem using Kolmogorov complexity and Grassmannian tools.

Contribution

It introduces a Kaufman-type exceptional set estimate for sets with optimal oracles and generalizes Marstrand's projection theorem conditions.

Findings

Proves a new exceptional set estimate for sets with optimal oracles.

Generalizes Marstrand's projection theorem to broader classes of sets.

Develops new tools based on Kolmogorov complexity for Grassmannian analysis.

Abstract

We prove a Kaufman-type exceptional set estimate for sets in $\mathbb{R}^n$ that have optimal oracles, a class of sets that strictly contains the analytic sets and sets with equal Hausdorff and packing dimension. As a consequence, we generalize the conditions under which Marstrand's projection theorem for $k$-planes is known to hold. Our proofs use effective methods, especially Kolmogorov complexity, and along the way, we introduce several new tools for studying the information content of elements of the Grassmannian.