# Improving the bounds for Roth's theorem

### From Polymath1Wiki

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.

## Contents |

## Background material

Sketch of the Roth/Meshulam argument for cap sets

Sketch of Bourgain's first argument using Bohr sets

## The Bateman-Katz argument

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

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.

- Tom Sanders, On Roth's theorem on progressions. The paper that gets within loglog factors of the 1/logN density barrier in Roth's theorem.

- Ernie Croot and Olof Sisask, A probabilistic technique for finding almost-periods of convolutions. The paper that introduced a technique that is central to Sanders's proof.

- 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

- Blog post by Tom Sanders about the ideas and difficulties involved (Main research thread)

- Polymath6: A is to B as C is to ??? (Main discussion thread)

- 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.