On sums of Szemerédi-Trotter sets

Shkredov, Ilya D.

doi:10.1134/s0081543815040185

Cited by 30 publications

(40 citation statements)

References 13 publications

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“…Again, it is necessary to check the conditions of Lemma , where C > 0 is an absolute constant. Indeed, by the previous arguments the inequality |A + λA| ≤ M|A| gives us the required bound for the thirdenergy and the common energy, see Lemma 23 and Lemma 25, and exactly such kind of statements are needed in the eigenvalues approach from [43]. Nevertheless, the methods in [43] are less elementary (although give slightly better bounds), so we prefer to give a more transparent proof in the current paper.…”

Section: Proof Of Theoremmentioning

confidence: 90%

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: 90%

New Results on Sum‐product Type Growth Over Fields

et al. 2019

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“…Dealing with the "few sums, many products" side of the coin has been much more successful; this was first done by Elekes and Ruzsa [9]. Its counterpart proves to be much harder; it is referred by some authors as the weak Erdős-Szemerédi conjecture, with the best known estimate stated as [17,Theorem 12], originating in [24].…”

Section: 12mentioning

confidence: 99%

On the energy variant of the sum-product conjecture

2019

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We prove new exponents for the energy version of the Erdős-Szemerédi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard formulation of the question, both for general fields and the special case of real or complex numbers, and appear to be the best ones attainable within the currently available technology. Further results are obtained about multiplicative energies of additive shifts and a strengthened energy version of the "few sums, many products" inequality of Elekes and Ruzsa. The latter inequality enables us to obtain a minor improvement of the state-of the art sum-product exponent over the reals due to Konyagin and the second author, up to 4 3 + 1 1509 . An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented.

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“…Improvements have been made to (2), which highlights the advantage of allowing B to vary in Definition 1.5. Sh1,Theorem 11] , [KoRu,Corollary 10], see also [ScSh], [MRS2,Theorem 13], see also [Sh3]] Let A ⊂ R. Then…”

Section: Definition 15 Letmentioning

confidence: 99%

“…Solymosi's argument with Shkredov's work in additive combinatorics [Sh1,Sh3] to increase Solymosi's exponent. In a more recent paper [KoSh2], the same authors proved δ = 4/3 + 5/9813 is admissible in Conjecture 1.1.…”

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confidence: 99%

On higher energy decompositions and the sum–product phenomenon

Shakan¹

2018

Math. Proc. Camb. Phil. Soc.

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Let A ⊂ R be finite. We quantitatively improve the Balog-Wooley decomposition, that is A can be partitioned into sets B and C such thatWe use similar decompositions to improve upon various sum-product estimates. For instance, we show |A + A| + |AA| |A| 4/3+5/5277 . The author is partially supported by NSF grant DMS-1501982 and would like to thank KevinFord for financial support. The author also thanks Kevin Ford and Oliver Roche-Newton for useful comments and suggestions, as well as the referee for a meticulous and timely reading and helpful suggestions.

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On sums of Szemerédi-Trotter sets

Abstract: We prove new general results on sumsets and difference sets for sets of the Szemerédi-Trotter type. This family includes convex sets, sets with small multiplicative doubling, images of sets under convex/concave maps and others.

Cited by 30 publications

References 13 publications

New Results on Sum‐product Type Growth Over Fields

New Results on Sum‐product Type Growth Over Fields

On the energy variant of the sum-product conjecture

On higher energy decompositions and the sum–product phenomenon

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