Hausdorff dimension of plane sections and general intersections

Abstract

This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. 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, $s>m$. As an application we prove for measurable sets $A,B\subset\mathbb{R}^{n}$ with positive $s$- and $t$-dimensional measures that if $s + (n-1)t/n > n$, then $\dim A\cap (g(B)+z) \geq s+t - n$ for almost all rotations $g$ and for positively many $z\in\mathbb{R}^{n}$. We shall also give an application on the estimates of the dimension of the set of exceptional rotations.