Additive structures in sumsets

Sanders, Tom

doi:10.1017/s030500410700093x

Cited by 31 publications

(40 citation statements)

References 7 publications

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“…In [San08] the result of Green and Tao is refined with Chang's theorem. One can use the same techniques to refine Proposition 7.2 with Theorem 2.2, our A(G)-analogue of Chang's theorem; doing so gives the following.…”

Section: A Structural Results For the Local Fourier Spectrummentioning

confidence: 99%

The Littlewood-Gowers problem

Sanders

2007

J Anal Math

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Abstract. The paper has two main parts. To begin with suppose that G is a compact Abelian group. Chang's Theorem can be viewed as a structural refinement of Bessel's inequality for functions f ∈ L 2 (G). We prove an analogous result for functions f ∈ A(G), where A(G) is the space {f ∈ L 1 (G) : f 1 < ∞} endowed with the norm f A(G) := f 1 , and generalize this to the approximate Fourier transform on Bohr sets.As an application of the first part of the paper we improve a recent result of Green and Konyagin: Suppose that p is a prime number and A ⊂ Z/pZ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that χ A A(Z/pZ) ≫ε (log p) 1/3−ε ; we improve this to χ A A(Z/pZ) ≫ε (log p) 1/2−ε . To put this in context it is easy to see that if A is an arithmetic progression then χ A A(Z/pZ) ≪ log p.

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Section: A Structural Results For the Local Fourier Spectrummentioning

confidence: 99%

The Littlewood-Gowers problem

Sanders

2007

J Anal Math

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“…These results terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/fms.2016.2 focus on the small density case: when α is large some prior results can offer better bounds; for example, for F n 2 Sanders showed in [21] that 2A contains 1 − of an affine subspace of codimension at most Cα −2 log(1/ ), and codimension at most Cα −1 log(1/ ) in [22]. However, it is far from clear where the truth lies for these results -not only in terms of the exponents on the logarithms but also in the qualitative differences between 3A and 4A.…”

Section: Discussionmentioning

confidence: 99%

“…See for example [7,15,21,22] for more background. From the perspective of the present paper it is illuminating to consider what is known in this setting for 2A, 3A and 4A, for which the best bounds known for large densities are all due to Sanders.…”

Section: Density Rangementioning

confidence: 99%

Roth’s Theorem for Four Variables and Additive Structures in Sums of Sparse Sets

Schoen

Sisask

2016

Forum of Mathematics, Sigma

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We show that if A ⊆ {1, . . . , N } does not contain any nontrivial solutions to the equationwhere c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A + A + A contain long arithmetic progressions if A ⊆ {1, . . . , N }, or high-dimensional affine subspaces if A ⊆ F n q , even if A has density of the shape above.

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“…for some absolute constant c > 0. This bound was improved by Green [18] using a different Fourier-analytic argument to the best bound that is currently known for highdensity sets, increasing the exponent 1/3 above to 1/2; a similar bound has since also been established by Sanders [36] using another Fourier-analytic technique. By contrast, our result yields somewhat shorter arithmetic progressions for high-density sets (where α and β are thought of as not depending on N) but is also able to deal with sets that are much smaller than previously possible.…”

Section: Proposition 13 (Lmentioning

confidence: 95%

A Probabilistic Technique for Finding Almost-Periods of Convolutions

2010

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Abstract. We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A + B that works with sparser subsets of {1, . . . , N } than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A 1 · A 2 · A 3 and A 2 · A −2 are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in A · B.Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.

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Additive structures in sumsets

Cited by 31 publications

References 7 publications

The Littlewood-Gowers problem

The Littlewood-Gowers problem

Roth’s Theorem for Four Variables and Additive Structures in Sums of Sparse Sets

A Probabilistic Technique for Finding Almost-Periods of Convolutions

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