Hausdorff dimension of intersections with planes and general sets

We give conditions on a general family $P_λ:\R^n\to\R^m, λ\in Λ,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_λ^{-1}\{u\}=s-m$ 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 $z\in\Rn$.

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