Old course notes.
Restriction and Kakeya phenomena (Cambridge, 2002: quite out of date in many places)
Additive Combinatorics (Cambridge, 2009)
Ergodic theory (16 lecture 4th year course, Oxford 2014 and 2015, includes appendices on measure theory and basic Fourier analysis)
Analytic number theory (16 lecture 4th year course, Oxford 2016. Full proof of the prime number theorem and explicit formula.)
Riemann Integration (8 lectures, Oxford 2017)
Introduction to number theory (8 lectures, Oxford 2017)
- Chapter 1 (Roth's Theorem)
- Chapter 2 (Sumsets)
- Chapter 3 (Freiman-Ruzsa theorem)
- Chapter 4 (Balog-Szemeredi-Gowers)
- Chapter 5 (Longer progressions, Gowers norms)
- Chapter 6 (Sum-product over Fp, exponential sums over subgroups) became these notes.
- Chapter 7 (Sum-product in C)
Exercise sheets for the course: Sheet 1, Sheet 2, Sheet 3. Thanks to Sarah Penington for providing me with scanned and annotated copies of the old sheets, which I had lost.
Here are some odd bits and pieces that haven't found their way into published papers, blogs or the bin. Some of these were written over 15 years ago and haven't been updated since, so use at your own risk. Some are just quite standard bits of undergraduate material that I wrote up for one reason or another.
Edinburgh Lecture Notes on Freiman's Theorem (2002)
Progressions and Convex
Geometry (2005, supplement to the above)
Bourgain's bound for 3-term progressions (1999)
Bourgain's work on arithmetic progressions in sumsets (2000)
Heath-Brown-Szemeredi bound for 3-term progressions (1999)
is exceptionally close to an integer (1998).
An example proving that dense Freiman models of sets with small doubling need not exist in the nonabelian case (2007)
The Hahn-Banach theorem (2011)
A couple of standard facts about continued fractions (2012)
Runge's theorem (2012)
The Baire Category Theorem (2009)
Review of Additive Combinatorics by Tao and Vu (2007)
A very brief review of measure theory (2008)
A quick primer on conditional expectation (2008)
The Asymmetric Balog-Szemeredi-Gowers theorem (2016)