New results on sums and products in ℝ

Konyagin, Sergeĭ; Shkredov, Ilya D.

doi:10.1134/s0081543816060055

Cited by 44 publications

(84 citation statements)

References 12 publications

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“…A refinement of the proof of (1.2) by the same authors [8] resulted in an improved exponent, and this was improved further in [11] to 4 3 + 1 1509 − o (1), which stands as the best estimate for the sum-product problem over real numbers at the time of writing. See [7], [8] and the references contained therein for more background on the sum-product problem.…”

Section: Introductionmentioning

confidence: 97%

“…Solymosi notably used a beautiful and elementary geometric argument to prove that, for any finite set

A \subset R

trueprefixmax false{false| A + A false|, false| A A false| false} ≫ \frac{{false| A false|}^{4 / 3}}{{prefixlog}^{1 / 3} false| A false|} .

Recently, a breakthrough for this problem was achieved by Konyagin and Shkredov . They adapted and refined the approach of Solymosi, whilst also utilising several other tools from additive combinatorics and discrete geometry, in order to prove that

trueprefixmax false{false| A + A false|, false| A A false| false} ≫ {| A |}^{\frac{4}{3} + \frac{1}{20598} - o false(1 false)} .

A refinement of the proof of by the same authors resulted in an improved exponent, and this was improved further in to

\frac{4}{3} + \frac{1}{1509} - o false(1 false)

, which stands as the best estimate for the sum–product problem over real numbers at the time of writing. See , and the references contained therein for more background on the sum–product problem.…”

Section: Introductionmentioning

confidence: 99%

“…They adapted and refined the approach of Solymosi, whilst also utilising several other tools from additive combinatorics and discrete geometry, in order to prove that

trueprefixmax false{false| A + A false|, false| A A false| false} ≫ {| A |}^{\frac{4}{3} + \frac{1}{20598} - o false(1 false)} .

A refinement of the proof of by the same authors resulted in an improved exponent, and this was improved further in to

\frac{4}{3} + \frac{1}{1509} - o false(1 false)

Section: Introductionmentioning

confidence: 99%

“…The classical tool in additive combinatorics for this situation is the Balog–Szemerédi–Gowers Theorem, which says that if

E_{+} false(A false)

is large then

A

contains a large subset with small sum set. However, recent progress, particularly in [, , ], has led to the development of different tools which are more effective than the Balog–Szemerédi–Gowers Theorem in the sum–product setting. In particular we will use the following result, which is [, Theorem 12].…”

Section: Introductionmentioning

confidence: 99%

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On the size of the set AA+A

Roche-Newton

Ruzsa

Shen

et al. 2018

J. London Math. Soc.

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Section: Introductionmentioning

confidence: 97%

“…Solymosi notably used a beautiful and elementary geometric argument to prove that, for any finite set

A \subset R

trueprefixmax false{false| A + A false|, false| A A false| false} ≫ \frac{{false| A false|}^{4 / 3}}{{prefixlog}^{1 / 3} false| A false|} .

trueprefixmax false{false| A + A false|, false| A A false| false} ≫ {| A |}^{\frac{4}{3} + \frac{1}{20598} - o false(1 false)} .

A refinement of the proof of by the same authors resulted in an improved exponent, and this was improved further in to

\frac{4}{3} + \frac{1}{1509} - o false(1 false)

Section: Introductionmentioning

confidence: 99%

“…They adapted and refined the approach of Solymosi, whilst also utilising several other tools from additive combinatorics and discrete geometry, in order to prove that

trueprefixmax false{false| A + A false|, false| A A false| false} ≫ {| A |}^{\frac{4}{3} + \frac{1}{20598} - o false(1 false)} .

A refinement of the proof of by the same authors resulted in an improved exponent, and this was improved further in to

\frac{4}{3} + \frac{1}{1509} - o false(1 false)

Section: Introductionmentioning

confidence: 99%

“…The classical tool in additive combinatorics for this situation is the Balog–Szemerédi–Gowers Theorem, which says that if

E_{+} false(A false)

is large then

A

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the size of the set AA+A

Roche-Newton

Ruzsa

Shen

et al. 2018

J. London Math. Soc.

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“…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).…”

mentioning

confidence: 99%

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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