Short Intensive course

Course Title:

Introduction to Metric Diophantine Approximation (AP19-1)

Resource Person:

Abstract:
The set of real numbers is uncountable, and hence to large for us to study each individual number one at a time. Consequently, in order to understand the real numbers, we often group them in classes according to their approximation properties by elements of a countable sets such as the rational numbers or the algebraic numbers. The study of such approximations is known as Diophantine approximation. A natural question in this context is whether numbers enjoying certain prescribed properties exist. In some cases, it is possible to construct a number with some given properties explicitly; for instance in terms of its decimal expansion, its continued fraction expansion, as a limit of some explicit series etc. However, this is often rather difficult! It is often easier to use measure theoretical tools to show that a set of numbers enjoying the prescribed properties has non-zero measure (for some measure) and hence is non-empty. These tools will also allow us to discriminate between the `size’ of such sets of numbers, as well as to determine which properties are typical for real numbers.

Prerequisites:

In this course, we will start with a short introduction to (non-metric) Diophantine approximation, where after we will proceed with the metric theory. We will cover the required measure theoretical framework, including the required elements of fractal geometry, which is an essential tool in the study of metric Diophantine approximation.

Schedule:

Staring from August 28, 2019
Wednesday 14:00 – 15:30 and Friday 14:30 – 16:00

Venue: ASSMS Main Hall.

For registration and further details, please contact at,info@sms.edu.pk