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

Mon, 04/03/2013 - 14:00

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.