Topic
Sumset
About: Sumset is a research topic. Over the lifetime, 399 publications have been published within this topic receiving 4655 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, the authors combine I.G. Ruzsa's method and Y. Bilu's proof of Freiman's theorem to show that A is contained in a d-dimensional progression P, where P ≤ d(α) (0.3) and α ≤ c(α).
Abstract: .Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s theorem.A fundamental result in the theory of set addition is Freiman’s theorem. Let A ⊂Z be a finite set of integers with small sumset; thus assume|A + A| <α|A|, (0.1)whereA + A = {x + y |x,y ∈ A} (0.2)and | · | denotes the cardinality. The factor αshould be thought of as a (possiblylarge) constant. Then Freiman’s theorem states that A is contained in a d-dimensionalprogression P, whered ≤ d(α) (0.3)and|P||A|≤ C(α). (0.4)(Precise definitions are given in Section 1.) Although this statement is very intuitive,there is no simple proof so far, and it is one of the deep results in additive numbertheory.G. Freiman’s book [Fr] on the subject is not easy to read, which perhaps explainswhy in earlier years the result did not get its deserved publicity. More recently, two
271 citations
1 Jan 2005
TL;DR: The sparse regularity lemma and its applications Stefanie Gerke and Angelika Steger as discussed by the authors were the first to apply the sparse norm lemma to additive combinatorics.
Abstract: 1. Finite field models in additive combinatorics Ben Green 2. The subgroup structure of finite classical groups in terms of geometric configurations Oliver H. King 3. Constructing combinatorial objects via cliques Patric R. J. Ostergard 4. Flocks of circle planes Tim Penttila 5. Judicious partitions and related problems Alex Scott 6. An isoperimetric method for the small sumset problem O. Serra 7. The structure of claw-free graphs Maria Chudnovsky and Paul Seymour 8. The multivariate Tutte polynomial (alias Potts model) for graphs and matroids Alan D. Sokal 9. The sparse regularity lemma and its applications Stefanie Gerke and Angelika Steger.
205 citations
TL;DR: In this article, it was shown that the Shannon entropy of a discrete random variable X taking values in G is a function of the cardinality |A| of the finite set A ⊂ G.
Abstract: Let G = (G, +) be an additive group. The sumset theory of Plunnecke and Ruzsa gives several relations between the size of sumsets A + B of finite sets A, B, and related objects such as iterated sumsets kA and difference sets A − B, while the inverse sumset theory of Freiman, Ruzsa, and others characterizes those finite sets A for which A + A is small. In this paper we establish analogous results in which the finite set A ⊂ G is replaced by a discrete random variable X taking values in G, and the cardinality |A| is replaced by the Shannon entropy H(X). In particular, we classify those random variables X which have small doubling in the sense that H(X1 + X2) = H(X) + O(1) when X1, X2 are independent copies of X, by showing that they factorize as X = U + Z, where U is uniformly distributed on a coset progression of bounded rank, and H(Z) = O(1).
When G is torsion-free, we also establish the sharp lower bound $\Ent(X+X) \geq \Ent(X) + \frac{1}{2} \log 2-o(1)$ , where o(1) goes to zero as H(X) → ∞.
116 citations
TL;DR: For the problem of finding long arithmetic progressions in a sum-set, the authors showed that if the sum set contains no non-trivial three-term progressions, then |𝐴|≪𝑁(log log ǫ)4/log à 0.
Abstract: We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if 𝐴⊂{1,...,𝑁} contains no non-trivial three-term arithmetic progressions, then |𝐴|≪𝑁(loglog𝑁)4/log𝑁 . By the same method, we also improve the bounds in the analogous problem over 𝔽𝑞[𝑡] and for the problem of finding long arithmetic progressions in a sumset.
101 citations
TL;DR: A new method to bound the cardinality of product sets in groups and give three applications, including a new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made.
Abstract: We present a new method to bound the cardinality of product sets in groups and give three applications. A new and unexpectedly short proof of the Plunnecke-Ruzsa sumset inequalities for commutative groups. A new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made. A further generalisation of the Plunnecke-Ruzsa inequalities in general groups.
83 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 24 |
| 2020 | 27 |
| 2019 | 26 |
| 2018 | 25 |
| 2017 | 22 |