"Geometry of Fractal Sets from the Perspective of Fourier Analysis and Projection Theory"

Fractals are sets with intricate structure at infinitely many scales. One robust way to deal with such arbitrary objects is to decompose them into more usable components. Two powerful methods of decomposition include the Fourier transform and projection theorems. In this talk, we use these tools to establish relationships between the dimension (Hausdorff or Fourier dimension) of a set and the measure and interior of the images of the set under various operations, including sums, products, and distances. For instance, we consider lower bounds on the dimension of the product set $XY$, where $X$ is a set of scalars and $Y$ is a subset of Euclidean space. We determine the measure of $X+S^1$ when $X$ is a set of Hausdorff dimension $1$, as well as the interior of $X+S^1$ when $X$ is a suitable Cartesian product of Cantor sets.  We then use these result to study distance sets, and we give the first known result in the literature on the interior of pinned distance sets.

 

Host: Brett Wick

Tea will be served at 3:45 in room 200