Fourier analysis in combinatorial number theory

In this survey applications of harmonic analysis to combinatorial number theory are considered. Discussion topics include classical problems of additive combinatorics, colouring problems, higher-order Fourier analysis, theorems about sets of large trigonometric sums, results on estimates for trigonometric sums over subgroups, and the connection between combinatorial and analytic number theory. Bibliography: 162 titles.

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[80]  Jean Bourgain,et al.  Estimates on exponential sums related to the Diffie–Hellman Distributions , 2005 .

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[84]  An application of a local version of Chang's theorem , 2006, math/0607668.

[85]  Neil Hindman,et al.  Multiplicative structures in additively large sets , 2006, J. Comb. Theory, Ser. A.

[86]  József Solymosi,et al.  Arithmetic Progressions in Sets with Small Sumsets , 2005, Combinatorics, Probability and Computing.

[87]  Terence Tao.  A Quantitative Ergodic Theory Proof of Szemerédi's Theorem , 2006, Electron. J. Comb..

[88]  E. Szemerédi,et al.  Finite and infinite arithmetic progressions in sumsets , 2006 .

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[93]  Jean Bourgain,et al.  Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order , 2006 .

[94]  J. Bourgain,et al.  Exponential sum estimates over subgroups and almost subgroups of $$ \mathbb{Z}_{Q}^{*} $$, where Q is composite with few prime factors , 2006 .

[95]  Ben Green,et al.  Montreal Lecture Notes on Quadratic Fourier Analysis , 2006 .

[96]  Derrick Hart,et al.  Sums and products in finite fields: an integral geometric viewpoint , 2007, 0705.4256.

[97]  I. Shkredov.  On sumsets of dissociated sets , 2007, 0712.1074.

[98]  M. Z. Garaev,et al.  THE SUM-PRODUCT ESTIMATE FOR LARGE SUBSETS OF PRIME FIELDS , 2007, 0706.0702.

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[104]  J. Bourgain,et al.  Some Arithmetical Applications of the Sum-Product Theorems in Finite Fields , 2007 .

[105]  Mei-Chu Chang,et al.  On a question of Davenport and Lewis and new character sum bounds in finite fields , 2008 .

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[116]  Jean Bourgain,et al.  Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources , 2009 .

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[119]  Tim Austin.  On the norm convergence of non-conventional ergodic averages , 2008, Ergodic Theory and Dynamical Systems.

[120]  Ben Green,et al.  New bounds for Szemerédi's theorem, I: progressions of length 4 in finite field geometries , 2009 .

[121]  Norbert Hegyv'ari,et al.  Explicit constructions of extractors and expanders , 2012, 1206.1146.

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[126]  Terence Tao,et al.  The inverse conjecture for the Gowers norm over finite fields via the correspondence principle , 2008, 0810.5527.

[127]  József Solymosi,et al.  Sum-product Estimates in Finite Fields via Kloosterman Sums , 2010 .

[128]  M. Garaev,et al.  An Explicit Sum-Product Estimate in p , 2010 .