"The positive integers do not lie, like the logical foundations of mathematics, in the scarcely visible distance, nor in the uncomfortably tangled foreground, like the immediate data of the physical world, but a decent middle distance, where the outlines are clear and yet some element of mystery remains...There is no one so blind that he does not see them, and no one so sharp-sighted that his vision does not fail; they stand there a continual and inevitable challenge to the curiosity of every healthy mind. I have merely directed your attention for a moment to a few of the less immediately conspicuous features of the landscape, in the hope that I may sharpen your curiosity a little, and that some may feel tempted to walk a little nearer and take a closer view."

G. H. Hardy

This is the course page for the TCC course Density Increment Methods in Additive Combinatorics, 2024.

• There will be 8 lectures, Fridays from 10am-noon. These will be (technical issues permitting) delivered live in the TCC room in the Mathematical Institute, Oxford. Those from other universities part of the TCC course who wish to attend should join the Teams group, by emailing tcc@maths.ox.ac.uk with your Teams email address.
• The current lectures notes (updated as we go alone) are available here.
• Precisely what will be covered in each lecture has yet to be determined, since it depends on the speed of progress. Below is what was covered in past lectures and what plans to be covered in the next lecture:
  • Lecture One: Introduction to the density increment method; notation and preliminaries.
  • Lecture Two: The finite field model; Meshulam's theorem on 3APs in F_p^n.
  • Lecture Three: Bohr sets and Bourgain's bound for 3APs.