Improving the bounds for Roth's theorem: Difference between revisions
Added a bit more content to page |
Adding section headers for the BK argument; these may need to be revised as our understanding develops |
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[[Sketch of Bourgain's first argument using 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]] | |||
==Annotated bibliography== | ==Annotated bibliography== | ||
Revision as of 17:09, 5 February 2011
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
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
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.
