Additive combinatorics

@inproceedings{Tao2007AdditiveC,
  title={Additive combinatorics},
  author={Terence Tao and Van H. Vu},
  booktitle={Cambridge studies in advanced mathematics},
  year={2007}
}
  • T. Tao, V. Vu
  • Published in
    Cambridge studies in advanced…
    2007
  • Mathematics
2.4. Vinogradov’s three-primes theorem. Sound/Gowers spend some effort discussing Vinogradov’s result that all large odd integers are the sum of three primes. We will give their (and Hardy-Littlewood’s) proof, conditional on GRH. This introduces the circle method, which is a nice application of Fourier analytic techniques to additive problems. I also plan on one additional lecture on the circle method which briefly explains its other applications: Vinogradov without GRH, partitions, Waring’s… 
AN $L$ -FUNCTION-FREE PROOF OF VINOGRADOV’S THREE PRIMES THEOREM
  • X. Shao
  • Mathematics
    Forum of Mathematics, Sigma
  • 2014
Abstract We give a new proof of Vinogradov’s three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on
Plünnecke ’ s Inequality 2 . 1 Commutative graphs : definition and notation
Plünnecke’s inequality is the standard tool to obtain estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the
Bohr sets and multiplicative Diophantine approximation
  • Sam Chow
  • Mathematics
    Duke Mathematical Journal
  • 2018
In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem,
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This report contains a good part of the results of my reserach in additive combinatorics I have been conducting during the last decade, the central theme being the structural theory of set addition.
J. (2015). Finite field models in arithmetic combinatorics – ten years on. Finite Fields and Their Applications , 32 , 233-274.
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  • 2014
. It has been close to ten years since the publication of Green’s influential survey Finite field models in additive combinatorics [ ? ], in which the author championed the use of high-dimensional
Higher-rank Bohr sets and multiplicative diophantine approximation
Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich,
Affine linear sieve, expanders, and sum-product
AbstractLet $\mathcal{O}$ be an orbit in ℤn of a finitely generated subgroup Λ of GLn(ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial
On Additive Doubling and Energy
TLDR
It is shown that there is a universal $\epsilon>0$ so that any subset of an abelian group with subtractive doubling $K$ must be polynomially related to a set with additive energy at least $\frac{1}{K^{1-\ep silon}}$.
An introduction to additive combinatorics
This is a slightly expanded write-up of my three lectures at the Additive Combinatorics school. In the first lecture we introduce some of the basic material in Additive Combinatorics, and in the next
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