Intersections of sets and Fourier analysis

A classical theorem due to Mattila says that if A, B ⊂ ℝ d of Hausdorff dimension s A , s B respectively with s A + s B ≥ d, s B > (d + 1)/2, and dim H (A × B) = s A + s B ≥ d, then dimH(A∩(z+B)⩽sA+sB−d dimH(A∩(z+B)⩽sA+sB−d for almost every z ∈ ℝ d , in the sense of Lebesgue measure. In this paper, we replace the Hausdorff dimension on the left hand side of the first inequality above by the lower Minkowski dimension and replace the Lebesgue measure of the set of translates by a Hausdorff measure on a set of sufficiently large dimension. Interesting arithmetic issues arise in the consideration of sharpness examples