Cartesian Products Avoiding Patterns
Jacob Denson

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 with…
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Taxonomy
TopicsMathematical Dynamics and Fractals
