Large Sets Avoiding Rough Patterns

Abstract: The pattern avoidance problem seeks to construct a set $X\subset \mathbb{R}^d$ with large dimension that avoids a prescribed pattern. Examples of such patterns include three-term arithmetic progressions (solutions to $x_1 - 2x_2 + x_3 = 0$), or more general patterns of the form $f(x_1, \dots, x_n) = 0$. 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 necessarily given by the zero-set of a function with prescribed regularity. There are several problems that fit into the framework of rough pattern avoidance. As a first application, if $Y \subset \mathbb{R}^d$ is a set with Minkowski dimension $\alpha$, we construct a set $X$ with Hausdorff dimension $d-\alpha$ such that $X+X$ is disjoint from $Y$. As a second application, if $C$ is a Lipschitz curve, we construct a set $X \subset C$ of dimension $1/2$ that does not contain the vertices of an isosceles triangle.