Hausdorff dimension of intersections with planes and general sets

TL;DR

This paper establishes conditions under which the Hausdorff dimension of intersections with projections and translated sets behaves predictably, extending classical results to more general families of projections and transformations.

Contribution

It provides new criteria for the Hausdorff dimension formula to hold for general projection families and applies these to intersections of sets under rotations and translations.

Findings

Dimension formula holds generically for a broad class of projections.

Intersections of sets under rotations and translations have predictable Hausdorff dimension.

Results extend classical projection theorems to more general settings.

Abstract

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable sets $A\subset\Rn$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$, and with positive lower density. As an application we prove for measurable sets $A,B\subset\Rn$ with positive $s$- and $t$-dimensional measures, and with positive lower density that if $s + (n-1)t/n > n$, then $\dim A\cap (g(B)+z) = s+t - n$ for almost all rotations $g$ and for positively many $z\in\Rn$.