Date: Friday, February 09, 2018
Location: 3866 East Hall (3:00 PM to 4:00 PM)
Title: Interior, Dimension, and Measure of Algebraic Sums of Fractal Sets and Curves via Fourier analysis and other techniques
Abstract: It is a time honored and classic problem to ask for the properties of the algebraic sum $A+B$ given sets $A$ and $B$ in the Euclidean plane. We focus on the case when $Gamma$ is a piecewise $mathcal{C}^2$ curve (such as the unit circle). There is a natural guess what the size (Hausdorff dimension, Lebesgue measure) of $A+Gamma$ should be. We verify this under some natural assumptions. We also address the more difficult question: under which condition does the set $A+Gamma$ have nonempty interior? The results have some surprising consequences for distance sets: $$Delta_x(A) :={xy: y in A},$$ where $x$ is a fixed point and $A$ is a fractal subset of $mathbb{R}^d$ of sufficient Hausdorff dimension. The relation between structure within a fractal set (as measured by sufficient Hausdorff dimension or by the existence of geometric configurations within) and the Fourier decay of a measure supported on said set is implicit.
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Speaker: Krystal Taylor
Institution: OSU
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