Exploring Domains of Approximation in R: Expository Essay

In this paper I explore the concept of the Domains of Best Approximations. These structures are of interest in the study of Diophantine Approximation as they encode the inequalities that govern how efficiently an irrational point can be approximated by a rational point with a given bound on the (lowest common) denominator. The problem of best approximation can be formulated as follows. Given a rational point p ∈ Q, and certain constraints, what are all the points r ∈ R that point p best approximates. The best approximation is defined by constraints on a distance between p, r and other points of Q. Precise definition will be discussed below. In this paper I investigate the case when d = 2. Nosratieh has shown that there are points in Q for which the best approximation set is not convex. She gave a specific example of a point ( 17 33 , 8 33 ) which generates a non-convex domain of best approximation. One of the goals of this paper is to empirically find more points with non-convex domains of approximation. Of particular interest will be such points with the smallest denominator. Before considering the case d = 2, I looked at the concept of best approximation for d = 1, points in R.