Hausdorff dimension and projections related to intersections

For $S_g(x,y)=x-g(y),x,y\in {\mathbb{R}}^n,g\in O(n),$ we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given the dimension of $A$, both for general Borel subsets of ${\mathbb{R}}^{2n}$ and for product sets.