A classical theorem due to Mattila states that if $A,B \subset {\Bbb R}^d$ of Hausdorff dimension $s_A$ and $s_B$ respectively, such that $s_A+s_B \ge d$, $s_B>\frac{d+1}{2}$, and $dim_{{\mathcal H}}(A \times B)=s_A+s_B\ge d$, then

$$ dim_{{\mathcal H}}(A \cap (z+B)) \leq s_A+s_B-d$$ for almost every $z \in {\Bbb R}^d$ in the sense of Lebesgue measure.

We obtain a variant of this result in which we replace the Hausdorff dimension on the left-hand-side of the inequality above with the upper Minkowski dimension. We also provide an upper bound on the Hausdorff dimension of the set of translates which violate this inequality and give examples. This is joint work with Alex Iosevich and Suresh Eswarathasan. Techniques of harmonic analysis play a crucial role in our investigation.