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Generalizations of Fourier analysis, and how to apply them

Author: W. T. Gowers
Journal: Bull. Amer. Math. Soc. 54 (2017), 1-44
MSC (2010): Primary 05D99
Published electronically: September 14, 2016
MathSciNet review: 3584096
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Abstract: This is a survey of the use of Fourier analysis in additive combinatorics, with a particular focus on situations where it cannot be straightforwardly applied but needs to be generalized first. Sometimes very satisfactory generalizations exist, while sometimes we have to make do with theories that have some of the desirable properties of Fourier analysis but not all of them. In the latter case, there are intriguing hints that there may be more satisfactory theories yet to be discovered. This article grew out of the Colloquium Lectures at the Joint Meeting of the AMS and the MAA, given in Seattle in January 2016.

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W. T. Gowers
Affiliation: Department of Mathematics, University of Cambridge, DPMMS, 16 Mill Lane, Cambridge CB2 1SB, United Kingdom

Received by editor(s): July 31, 2016
Published electronically: September 14, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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