Higher moments of convolutions

Schoen, Tomasz; Shkredov, Ilya D.

doi:10.1016/j.jnt.2012.10.010

Cited by 83 publications

(134 citation statements)

References 26 publications

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“…Recall that this is defined as In particular, the upper bound has the advantage that it is essentially optimal in the case when |A ± A| is very small. Let us see how Lemma 23 can be used to prove Theorem 3, using the methods of [38] and [39]. The proof of Lemma 23 is given in Appendix B.…”

Section: Proof Of Theoremmentioning

confidence: 99%

New Results on Sum‐product Type Growth Over Fields

et al. 2019

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We prove a range of new sum-product type growth estimates over a general field F, in particular the special case F = F p . They are unified by the theme of "breaking the 3/2 threshold", epitomising the previous state of the art.This concerns two pivotal for the sum-product theory questions, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of F, as well as the number of values of the symplectic form determined by a finite subset of F 2 .We establish the estimate |R[A]| |A| 8/5 for cardinality of the set R[A] of distinct cross-ratios, defined by triples of elements of a (sufficiently small if F has positive characteristic, similarly for the rest of the estimates) set A ⊂ F, pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set P ⊂ F 2 , the number of distinct values of the symplectic form ω on pairs of distinct points u, u ′ of P is |ω(P )| |P | 2/3+c , with an explicit c. Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein.Our proofs strongly use specially suited applications of new incidence bounds over F, which apply to higher moments of representation functions. The technical thrust of the paper is using additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any F, by showing that if A is sufficiently small and has additive doubling constant M , then * |AA| M −2 |A| 14/9 . This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set A× A ⊂ F 2 , the quantity often arising in applications of geometric incidence estimates. (Expansion for large sets) IfF = F p , then |R[A]| ≫ min p, |A| 5/2 p 1/2 . In particular, if |A| ≥ p 3/5 , then |R[A]| ≫ p. 3. (Uniform expansion) If F = F p , then |R[A]| ≫ min{p, |A| 3 2 + 1 22 log − 4 9 |A|}.

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Section: Proof Of Theoremmentioning

confidence: 99%

New Results on Sum‐product Type Growth Over Fields

et al. 2019

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“…Следующая лемма показывается, что "старшие суммы" могут быть выражены в терминах обычных сумм (см., например, [10]). …”

Section: предварительные результатыunclassified

Number Theory and Applications, 2

Шкредов¹,

Shkredov²

2017

Sovrem. Probl. Mat.

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“…Using Definition 5 and easy calculations, one can obtain upper bounds for some simple characteristics of SzT-type sets (see, e.g., [8,9,12,13]). It is more convenient to use c (A, B) instead of the parameter α in the following lemma.…”

Section: Letmentioning

confidence: 99%