On higher energy decompositions and the sum–product phenomenon

Shakan, George

doi:10.1017/s0305004118000506

Cited by 24 publications

(41 citation statements)

References 33 publications

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“…We will use δ 1 to denote the best constant for which we know (1.1) to hold. The current record rests with Shakan [8], who showed that δ 1 = 1/3 + 5/5277 is permissible. Our main result extends the sum-product phenomenon to sets in…”

Section: Introductionmentioning

confidence: 61%

“…The sumproduct conjecture was first posed by Erdős and Szemerédi in [5]. Since then, numerous authors have worked on estimates of the form (1.1) in the case where A is either a subset of real numbers or some finite field (see [6] and [8]). In recent years, considerable work has also been done to extend these bounds to other rings and fields.…”

Section: Introductionmentioning

confidence: 99%

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Sum-Product Estimates for Diagonal Matrices

Mudgal

2020

Bull. Aust. Math. Soc.

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Given $d\in \mathbb{N}$ , we establish sum-product estimates for finite, nonempty subsets of $\mathbb{R}^{d}$ . This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, nonempty set of $d\times d$ diagonal matrices with real entries. Then, for all $\unicode[STIX]{x1D6FF}_{1}<1/3+5/5277$ , $$\begin{eqnarray}|A+A|+|A\cdot A|\gg _{d}|A|^{1+\unicode[STIX]{x1D6FF}_{1}/d},\end{eqnarray}$$ which strengthens a result of Chang [‘Additive and multiplicative structure in matrix spaces’, Combin. Probab. Comput.16(2) (2007), 219–238] in this setting.

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Section: Introductionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Sum-Product Estimates for Diagonal Matrices

Mudgal

2020

Bull. Aust. Math. Soc.

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confidence: 70%

An Improved Sum-Product Bound for Quaternions

Basit¹,

Lund²

2019

SIAM J. Discrete Math.

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We show that there exists an absolute constant c > 0, such that, for any finite set A of quaternions, max{|A + A, |AA|} |A| 4/3+c .This generalizes a sum-product bound for real numbers proved progress was made by Konyagin and Shkredov [9], by Rudnev, Shkredov and Stevens [12], and by Shakan [13]. Currently, Shakan's result gives the best bound for δ, showing that Conjecture 1.1 holds with δ ≤ 1/3 + 5/5277, whenever A is a set of real numbers. The conjecture has also been studied for other fields and rings. Konyagin and Rudnev [7] generalized Solymosi's geometric argument to finite sets of complex numbers, showing that Conjecture 1.1 holds with δ ≤ 1/3 − ε for any ε > 0 whenever A ⊂ C. For quaternions, Chang [1] proved the bound δ ≤ 1/54. This was improved upon by Solymosi and Wong [16], who showed that, for finite subsets of the quaternions, Conjecture 1.1 holds with δ ≤ 1/3 − ε, for any ε > 0. For more detail on the sum-product conjecture and its variants, see the recent survey of Granville and Solymosi [6].Our contribution is to generalize Konyagin and Shkredov's proof to quaternions, hence passing the δ = 1/3 barrier.Theorem 1.2. Let A be a finite set of quaternions. Then, there is a constant c > 0 such thatWe make no attempt to prove Theorem 1.2 for the largest possible value of c, instead preferring to keep the exposition relatively simple and self contained.Overview Our proof follows the general outline of Konyagin-Shkredov in [8] and [9]. Since our aim is to keep this paper self contained rather than obtain the best value for c, at various points we make do with weaker estimates than the ones used in these papers.We split the problem into two cases, depending on the additive energy of A (see Section 2 for the definitions). In the case that this additive energy is small, we prove an appropriate generalization of Solymosi's argument, in the spirit of and Solymosi-Wong [16] (see Section 4.3). In the case that this additive energy is large, we adapt the arguments of Konyagin and Shrkedov [9] to work for quaternions (see Section 4.2). This requires us to replace an application of the Szmerédi-Trotter theorem by a generalization proved by Solymosi and Tao [15], and to adapt the definitions and arguments of Konyagin and Shrkedov to work when multiplication is not commutative. This is done in Section 3. PreliminariesGiven finite subsets A, B of the quaternions, H, the sum set and product set are defined respectively asand AB := {ab : a ∈ A, b ∈ A}.We define the negation of A to be −A := {−a : a ∈ A}, and the inverse of A to be A −1 := {a −1 : a ∈ A}.The difference set is defined to be A − B, and the ratio set is defined as A/B := AB −1 ∩ B −1 A.

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“…(1.4) Many authors have worked on this kind of problem, and in particular, Solymosi [13] proved that one can take δ < 1/3 in (1.4). The current best known bound was given by Shakan [12] which states that δ < 1/3 + 5/5277 is permissible in (1.4).…”

Section: Introductionmentioning

confidence: 99%