Sums of Reciprocals and the Three Distance Theorem

This thesis is primarily concerned with sums of reciprocals in number theory. We first give a brief background and introduction to metric number theory, in particular the areas of Diophantine approximation and continued fractions. We then review sums of reciprocals and their significance and usefulness in certain fields of mathematics as a motivation for their study. The Three Distance theorem, also known as the Steinhaus conjecture, is then discussed and we use it to develop a new technique for obtaining bounds for sums of reciprocals, making constants explicit.