Krystal Taylor: Interior, Dimension, and Measure of Algebraic Sums of Planar Sets and Curves

Recently considerable attention has been given to the study of the arithmetic sums of two planar sets A + Г := {a + g : a ∈ A, g ∈ Г} . We focus on the case when Г is a piecewise C² curve, in particular when Г is the unit circle. There is a natural guess what the size (Hausdorff dimension, Lebesgue measure) of A + Г should be. We verify this under some simple natural assumptions. We also address the more difficult question: under which condition does the set A + Г have non-empty interior? Particular emphasis is given to the case when the Hausdorff dimension of the set A is 1. This is joint work with Karoly Simon.