Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes

TL;DR

This paper establishes strong versions of Marstrand's projection theorems, demonstrating that for certain sets, the typical projections preserve Hausdorff dimension, and identifies exceptional directions with Lebesgue measure zero.

Contribution

The paper introduces stronger forms of Marstrand's theorems, showing the independence of exceptional directions from subsets and applying duality to analyze intersections with hyperplanes.

Findings

Exceptional directions form a measure-zero set.

Projections of positive measure subsets preserve Hausdorff dimension.

Results apply to intersections with families of lines or hyperplanes.

Abstract

We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such that the projection onto any line with direction outside X, of any subset F of E of positive s-dimensional measure, has Hausdorff dimension min(1,s), i.e. the set of exceptional directions is independent of F. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.