Cartesian Products Avoiding Patterns

TL;DR

This paper explores methods to construct fractal sets that avoid specific patterns, including polynomial and rough patterns, with applications to sumsets and geometric configurations, expanding on previous work and introducing new strategies.

Contribution

It introduces new approaches to avoid rough patterns in fractal sets and provides results on pattern avoidance in sumsets and geometric configurations.

Findings

Existence of sets Y with complementary dimensions avoiding sumset intersections with X

Construction of 1/2-dimensional subsets of Lipschitz curves avoiding isosceles triangle vertices

Extension of previous pattern avoidance methods to rough pattern scenarios

Abstract

The pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns, such as finding a set whose Cartesian product avoids the zero set of a given function. Previous work on the subject has considered patterns described by polynomials, or functions satisfying certain regularity conditions. We provide an exposition of some results in this setting, as well as considering new strategies to avoid what we call `rough patterns'. This thesis contains an expanded description of a method described in a previous paper by the author and his collaborators Malabika Pramanik and Joshua Zahl, as well as new results in the rough pattern avoidance setting. There are several problems that fit into the pattern of rough pattern avoidance. For instance, we prove that for any set $X$ with lower Minkowski dimension $s$, there exists a set $Y$ with Hausdorff dimension $1-s$ such that for any rational numbers $a_1, \dots, a_N$, the set $a_1 Y + \dots + a_N Y$ is disjoint from $X$, or intersects with $X$ solely at the origin. As a second application, we construct subsets of Lipschitz curves with dimension $1/2$ not containing the vertices of any isosceles triangle.