# EP/G050198/1 - Quadratic Fourier analysis in combinatorics.

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Principal Investigator - Sch of Mathematical Sciences, Queen Mary, University of London

Start Date

10/2009

End Date

09/2012

Value

£212,178

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One of the guiding principles of additive combinatorics is that if one has a highly structured mathematical object, say an abelian group, then one has to do a lot of work in order to remove certain types of structure from that object. A fundamental example of this principle is given by the celebrated theorem of Klaus Roth, dating back to 1953, stating that if one selects a positive proportion of the numbers between 1 and N, then one is forced to inherit some of the additive structure of this collection of numbers, provided N is large. In this case, the additive structure comes in the form of solutions to translation-invariant linear equations, giving rise to such structures as three-term arithmetic progressions x, x+d, x+2d where d is non-zero. In demonstrating this, Roth solved a 20-year old conjecture, yet it was to be another 20 years before a further major breakthrough was made along these lines: in 1975, Endre Szemeredi established that, in fact, any subcollection of positive density of the numbers between 1 and N must contain a very long arithmetic progression, not just one of three terms. However, Szemeredi's proof of this fact was completely combinatorial whereas Roth's used the theory of Fourier analysis, which led to a substantially more efficient argument. The question as to whether Roth's general method of proof could be extended to prove Szemeredi's result was therefore natural, and this was affirmatively answered in foundational work of Tim Gowers between 1998 and 2001, leading to the birth of so-called quadratic Fourier analysis. In the meantime, however, the (linear) Fourier analysis that Roth had used had been significantly developed and better understood, leading to many extensions of the subject. The main aim of this project is thus to try to develop quadratic Fourier analysis in a similar way to what has been possible with linear Fourier analysis, as well as to investigate links between modern additive Fourier analysis and recent extensions of the general method of Szemeredi's proof.

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Name: Quadratic Fourier analysis in combinatorics. - EP/G050198/1

Start Date: 2009-10-01T00:00:00+00:00

End Date: 2012-09-30T00:00:00+00:00

Organization: Queen Mary, University of London

Description: One of the guiding principles of additive combinatorics is that if one has a highly structured mathematical object, say an abelian group, then one has to do a lot of work in order to remove certain types of structure from that object. A fundamental example ...