modules

 

At its core, Diophantine Approximation is the branch of Number Theory concerned with understanding how well we can approximate real numbers by rational numbers. For example, we know that the rationals are dense in the reals, so given any real number we can find a rational arbitrarily close to that real number. However, what about if we start imposing restrictions on, say, the denominator of the rationals we want to find within a given neighbourhood of a particular real number? More generally, Diophantine Approximation is the study of such questions and aims to quantify more precisely how well we can approximate real numbers by rationals. 

 

Metric Number Theory is concerned with measure theoretic properties of sets of numbers (or points in higher dimensions) satisfying certain number theoretic properties. The sets we encounter in Diophantine Approximation lend themselves very naturally to being studied from a measure theoretic viewpoint. In this course, we will study some of the fundamental results and techniques in Diophantine Approximation, paying particular attention to the measure theoretic aspects, and also aim to highlight natural connections with other areas of mathematics such as Fractal Geometry, Dynamical Systems, and Ergodic Theory.

 

Pre-requisites:

Essential: MTH2008 Real Analysis

 

This module is especially recommended to those who enjoyed MTH3040 Topology and Metric Spaces and/or MTH3004 Number Theory.