# Intersections of fractal sets, multi-linear operators, and Fourier analysis

Mon, 04/03/2013 - 14:00
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Abstract:

A classical theorem due to Mattila says that if $A,B \subset {\Bbb R}^d$ of Hausdorff dimension $s_A, s_B$, respectively,
with $s_A+s_B \ge d$, $s_B>\frac{d+1}{2}$ 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 variable coefficient variant of this result in which we
are able to replace the Hausdorff dimension with the upper Minkowski
dimension on the left-hand-side of the first inequality.  This is
joint work with Alex Iosevich and Suresh Eswarathasan.  Fourier
Integral Operator bounds and other techniques of harmonic analysis
play a crucial role in our investigation.