Alexander Henderson Department of Mathematics and Statistics University of Nevada, Reno Assouad Dimension and the Open Set Condition Abstract: The Assouad dimension is a measure of the complexity of a fractal set that is of both practical and theoretical interest. For example, infinite dimensional dynamical systems with attractors A such that the set of differences A - A has finite Assouad dimension allow embeddings into finite dimensional spaces without losing the original dynamics. In this thesis, we generalize Moran's open set condition and introduce a notion called grid like which allows us to compute upper bounds and exact values for the Assouad dimension certain fractal sets that arise as the attractors of self-similar iterated function systems. Then for a fractal set A, we explore the question of whether the Assouad dimension of the set of differences A - A obeys any bound related to the Assouad dimension of A. We find that even in very simple, natural examples, such a bound does not generally hold. This result demonstrates how a natural phenomenon with a simple underlying structure has the potential to be difficult to measure.