University of British Columbia
Talk Title: Cartesian Products Avoiding Rough Patterns
Abstract: The pattern avoidance problem seeks to construct a subset of Euclidean space with large dimension that avoids a prescribed pattern such as collinear points, or the presence of the vertices of an isosceles triangles. Previous work on the subject has considered patterns described by polynomials, or by more general smooth functions satisfying certain regularity conditions. We consider the case of ‘rough patterns, i.e. a pattern specified with a set with fractional dimension. There are several problems that fit into the framework of rough pattern avoidance. As a first application, we construct sum-sets avoiding `rough’ sets. As a second application, given a set of dimension close to one, we can construct a subset of dimension 1/2 avoiding isosceles triangles.