Course Materials
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- Lecturer: Profile: James Maynard
Course information
General prerequisites:
Very little is required in terms of prerequisites; we will develop what we need from scratch. Basic ideas of complex analysis, elementary number theory and some familiarity with Fourier series will be helpful but not essential. ASO Number Theory is recommended.
Course term: Hilary
Course lecture information: 16 lectures
Course weight: 1
Course level: M
Assessment type: Written Examination
Course overview:
The aim is to study classical additive questions in number theory, particularly using ideas from combinatorics and Fourier analysis. Highlights of the course will include the classical theorem of Legendre that every whole number is the sum of four squares, results on Waring's problem showing that every number is the sum of s perfect kth powers (where s is a function of k), and Roth's theorem showing that every positive density subset of the integers contains infinitely many 3-term arithmetic progressions.
Course synopsis:
Sums of squares: Every prime congruent to 1 modulo 4 is the sum of two squares, and the classification of integers expressible as the sum of two squares. Every number is the sum of four squares. Discussion of sums of 3 squares and alternative methods of proof.
Fourier analysis for additive problems: Basic Fourier analysis. Solutions to additive equations modulo p in pseudorandom sets. Bounds for the Fourier transform of kth powers. Waring's problem modulo q.
Hardy-Littlewood circle method: Overview of the method. Minor arcs for Waring's problem. Major arcs for Waring's problem. The singular series.
Roth's Theorem: The density increment strategy, self-similarity.
The aim is to emphasise common universal themes even as the applications become more complicated.
Fourier analysis for additive problems: Basic Fourier analysis. Solutions to additive equations modulo p in pseudorandom sets. Bounds for the Fourier transform of kth powers. Waring's problem modulo q.
Hardy-Littlewood circle method: Overview of the method. Minor arcs for Waring's problem. Major arcs for Waring's problem. The singular series.
Roth's Theorem: The density increment strategy, self-similarity.
The aim is to emphasise common universal themes even as the applications become more complicated.
Section outline
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Full lecture notes for the course. Please note that earlier iterations of this course covered rather different content in places.
Please let me know of any comments or corrections.
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Covers material from Chapters 1 of the notes.
The sheet is available in two formats: with no solutions, and with solutions to Section A and C questions. -
Covers material from Section 3 and some of Section 4 of the notes.
The sheet is available in two formats: with no solutions, and with solutions to Section A and C questions.
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This sheet covers the second half of Chapter 4 and Chapter 5 of the notes.
It is available in two forms: without solutions and with solutions to Sections A and C.
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