Hausdorff dimension of plane sections and general intersections

This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family $P_{\lambda }:{\mathbb{R}}^n\to {\mathbb{R}}^m,\lambda \in \mathrm{\Lambda },$ 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 $s+(n-1)t\mathrm{/}n>n$, then $\dim A\cap (g(B)+z)\ge s+t-n$ 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.