Additive combinatorics

  title={Additive combinatorics},
  author={Terence Tao and Van H. Vu},
  booktitle={Cambridge studies in advanced mathematics},
  • T. Tao, V. Vu
  • Published in
    Cambridge studies in advanced…
  • 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… 
  • 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,
Algebraic and combinatorial methods in the theory of set addition
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.
  • Mathematics
  • 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
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