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MODULE TITLE	Metric Number Theory and Diophantine Approximation	CREDIT VALUE	15
MODULE CODE	MTHM014	MODULE CONVENER	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	33.00	Guided Independent Study	117.00	Placement / Study Abroad	0.00

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	20	Written Exams	80	Practical Exams	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

Reading list for this module:

Type	Author	Title	Publisher	Year	Search
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	[Library]
Set	Khincin, A. Y.	Continued Fractions			[Library]
Set	Schmidt, W. M.	Diophantine Approximation			[Library]
Set	Harman, G.	Metric Number Theory			[Library]
Set	Sprindžuk, V. G.	Metric Theory of Diophantine Approximations			[Library]
Set	Falconer, K.	Fractal Geometry: Mathematical Foundations and Applications			[Library]
Set	Billingsley, P.	Probability and Measure			[Library]

CREDIT VALUE	15	ECTS VALUE	7.5

PRE-REQUISITE MODULES	MTH2008
CO-REQUISITE MODULES

NQF LEVEL (FHEQ)	7	AVAILABLE AS DISTANCE LEARNING	No
ORIGIN DATE	Tuesday 17 January 2023	LAST REVISION DATE	Tuesday 17 January 2023

KEY WORDS SEARCH	Metric Number Theory, Diophantine Approximation, Khintchine's Theorem, Jarnik's Theorem, Continued Fractions, Lebesgue Measure, Hausdorff Measure

MTHM014 - Metric Number Theory and Diophantine Approximation (2023)