Improving the bounds for Roth's theorem

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This is a wiki for Polymath6, a project that aims to look at recent results of Bateman and Katz and of Sanders, and attempt to combine the ideas that go into them in order to break the log N-barrier in Roth's theorem. I hope that soon there will be expositions of their proofs, as well as of much of the background material needed to understand them. The wiki has only just been started, so has very little material on it so far.

Background material

Sketch of the Roth/Meshulam argument for cap sets

Basic facts about Bohr sets

Sketch of Bourgain's first argument using Bohr sets

The Bateman-Katz argument

Overview

Section 3: the "nd-estimate"

Section 4: the large spectrum contains many additive quadruples

Section 5: the large spectrum of a non-incrementing set is not additively smoothing

Section 6: the structure of additively non-smoothing sets

Section 7: the structure of the large spectrum of a non-incrementing cap set

Section 8: using the structure of the large spectrum to obtain density increments

Sanders's argument

Overview

The Croot-Sisask lemma

The Katz-Koester lemma

Sanders's local version of Chang's theorem

Annotated bibliography

  • Michael Bateman and Nets Katz, New bounds on cap sets. The paper that obtains an improvement to the Roth/Meshulam bound for cap sets.
  • Nets Katz and Paul Koester, On additive doubling and energy, Another paper that introduced a method that Sanders used and that is related to some of the ideas in the Bateman-Katz paper as well.

Links to more informal discussions of the problem

  • The way an argument might go, A document written by Nets Katz, summarizing discussions he had with Olof Sisask about the possibilities for combining the Bateman-Katz and Sanders papers.