Cartesian products avoiding patterns

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.