Szego Seminar: "Diophantine approximation and the Gauss transformation"

Speaker: Jeet Sampat, Washington University in Saint Louis

Abstract: In this introductory talk, we will discuss about the continued fraction representation of real numbers, and how they determine the “goodness/badness” of approximating a given real number x with rational numbers p/q. This ties in with the concept of Diophantine approximation, which will be explained briefly. For the second half, we shall discuss some topics in Ergodic theory. A lot of the results in the theory of Diophantine approximations can be proved using the Ergodicity of the Gauss transformation T(x) = 1/x – [1/x] on (0,1), where [.] is the floor function. We shall also see some interesting results about the growth of the partial quotients of continued fractions using the Gauss transformation.

Host: Nathan Wagner