Hausdorff dimension of intersections with planes and general sets

Abstract

We give conditions on a general family $P_{\lambda }:{\mathbb{R}}^n\to {\mathbb{R}}^m$, $\lambda \in \Lambda$, of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda }^{-1}\{u\}=s-m$ holds generically for measurable sets $A\subset {\mathbb{R}}^n$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$, and with positive lower density. As an application we prove for measurable sets $A,B\subset {\mathbb{R}}^n$ 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 $z\in {\mathbb{R}}^n$.