Large Sets Avoiding Rough Patterns

The pattern avoidance problem seeks to construct a set X ⊂ R d $$X\subset \operatorname {\mathrm {\mathbf {R}}}^d$$ with large dimension that avoids a prescribed pattern. Examples of such patterns include three-term arithmetic progressions (solutions to x 1 − 2x 2 + x 3 = 0), geometric structures such as simplices, or more general patterns of the form f(x 1, …, x n) = 0. Previous work on the subject has considered patterns described by polynomials or by functions f satisfying certain regularity conditions. We consider the case of “rough” patterns, not prescribed by functional zeros. There are several problems that fit into the framework of rough pattern avoidance. As a first application, if Y ⊂ R d $$Y \subset \operatorname {\mathrm {\mathbf {R}}}^d$$ is a set with Minkowski dimension α, we construct a set X with Hausdorff dimension d − α such that X + X is disjoint from Y . As a second application, if C is a Lipschitz curve with Lipschitz constant less than one, we construct a set X ⊂ C of dimension 1∕2 that does not contain the vertices of an isosceles triangle.