The theory of automatic sets and sequences arises naturally in many different areas of mathematics, notably in the study of algebraic power series in positive characteristic, due to work of Christol, and in Derksen's classification of zero sets for sequences satisfying a linear recurrence over fields of positive characteristic. A fundamental dichotomy for automatic sets shows that they are either sparse, having counting functions that grow relatively slowly, or they are not sparse, in which case their counting functions grow reasonably fast. While this dichotomy has been known to hold for some time, there has not---to this point in time---been a systematic study of the algebraic and number theoretic properties of sparse automatic sets.
This thesis rectifies this situation and gives multiple results dealing specifically with sparse automatic sets. In particular, we give a stronger version of a classical result of Cobham for automatic sets where one now specializes to sparse automatic sets; we then prove that a conjecture of Erdos and Turan holds for automatic sets, again using the theory of sparseness; finally, we give a refinement of a classical result of Christol where we consider algebraic power series whose support set is a sparse automatic set
