Inhomogeneous Diophantine Approximation

Diophantine approximation is a topic of number theory concerned with rational approximations of real, typically irrational, numbers. In other words, we seek q∈Z for a given α∈R such that ||qα|| (the distance from qα to the nearest integer) is small. This thesis serves as a primer on many of the famous results in this field. First, the fundamental result of Diophantine approximation, Dirichlet's theorem, is presented along with several methods of proof. Other theorems involving limits of accuracy are given before moving into inhomogeneous approximations (of the form ||qα- β|| for some β∈R). Finally, recent results in the inhomogeneous case are used to prove a theorem on irrational circle rotations with connections to ergodic theory.