Study information

# Metric Number Theory and Diophantine Approximation - 2023 entry

MODULE TITLE CREDIT VALUE Metric Number Theory and Diophantine Approximation 15 MTHM014 Dr Demi Allen (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
 Number of Students Taking Module (anticipated) 25
DESCRIPTION - summary of the module content

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.

AIMS - intentions of the module

The primary aim of this module will be to introduce you to the fields of Diophantine Approximation and Metric Number Theory. We will begin by studying basic properties of continued fractions, Dirichlet’s Theorem on approximation of real numbers, badly approximable numbers, and how all of these topics are tied together. We will then cover the basics of measure theory which will enable us to develop the theory of well-approximable sets. We will cover classical results of Khintchine (relating to Lebesgue measure) and Jarnik (relating to Hausdorff measure) in this area. We will introduce the Mass Transference Principle, a recently developed powerful tool in the fields of Diophantine Approximation and Metric Number Theory, which demonstrates a surprising connection between the Lebesgue measure theory and Hausdorff measure theory of sets we encounter in Diophantine Approximation. We will also aim to generalise the above mentioned theory to higher dimensions.

Time permitting, we may also delve into some more advanced topics which are at the forefront of research in these areas. In particular, topics we may consider include:
• Weighted Diophantine Approximation
• Diophantine Approximation on manifolds
• Diophantine Approximation in self-similar sets (i.e. fractals such as the Cantor set)

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

1. Demonstrate an understanding of the main concepts of Diophantine Approximation and be able to state some of the classical results in this area.
2. Be able to apply the techniques covered to be able to prove results and solve problems in this area.

Discipline Specific Skills and Knowledge

3. Recognise how Diophantine Approximation fits into the study of the broader topic of Number Theory and begin to develop an appreciation of how it is related to other areas of mathematics such as Fractal Geometry, Dynamical Systems, and Ergodic Theory.

Personal and Key Transferable / Employment Skills and Knowledge

4. Advanced problem solving skills and ability to communicate advanced material and proofs through written work.

SYLLABUS PLAN - summary of the structure and academic content of the module
• Continued fractions: definitions and basic properties
• Dirichlet’s Theorem and Hurwitz’s Theorem
• Badly approximable numbers, very well approximable numbers, and Liouville numbers
• Correspondence between badly approximable numbers and continued fractions
• Basics of Measure Theory (specifically the introduction of Lebesgue measure)
• Well-approximable points
• Khintchine’s Theorem
• Limitations of Lebesgue measure/Khintchine’s Theorem
• Hausdorff measure and dimension
• Jarník-Besicovitch Theorem
• Jarník’s Theorem
• The Mass Transference Principle
• Applications of the Mass Transference Principle
• Higher dimensional theory (generalisation of the classical results of Dirichlet, Khintchine, and Jarnik to higher dimensions)
Time permitting we may look at one of the following topics:
• Weighted Diophantine Approximation
• Diophantine Approximation on manifolds
• Diophantine Approximation in self-similar sets

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures (27 hours), Problems Classes (6 hours) Guided Independent Study 117 Private study, problems sheets, coursework, and assessment preparation.

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Submission of selected (un-assessed) problem sheet questions 5 hours All Annotated scripts/verbal

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework 1 – assessed problem sheet 10 15 hours All Annotated scripts/verbal
Coursework 2 – assessed problem sheet 10 15 hours All Annotated scripts/verbal
Written exam – closed book 80 2 hours All On request

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
All above Written examination (100%) All August Ref/Def period

RE-ASSESSMENT NOTES
Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 50%.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Type Author Title Edition Publisher Year ISBN
Set Beresnevich, V., F. Ramírez, S. Velani Metric Diophantine approximation: aspects of recent work. Dynamics and analytic number theory, 1–95, London Math. Soc. Lecture Note Ser., 437 CUP 2016
Set Khincin, A. Y. Continued Fractions
Set Schmidt, W. M. Diophantine Approximation
Set Harman, G. Metric Number Theory
Set Sprindžuk, V. G. Metric Theory of Diophantine Approximations
Set Falconer, K. Fractal Geometry: Mathematical Foundations and Applications
Set Billingsley, P. Probability and Measure
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH2008
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 7 No Tuesday 17th January 2023 Tuesday 17th January 2023
KEY WORDS SEARCH Metric Number Theory, Diophantine Approximation, Khintchine's Theorem, Jarnik's Theorem, Continued Fractions, Lebesgue Measure, Hausdorff Measure