An Improved Sum-Product Bound for Quaternions

Abstract

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.

Sum-product estimates for diagonal matrices

“…Arithmetic combinatorics, Sum-product estimates. 1 We note a recent improvement in this direction by Rudnev and Stevens who show that δ 1 < 1/3 + 2/1167 is permissible in (1.1), see preprint arXiv:2005.11145.…”

“…In recent years, considerable work has also been done to extend these bounds to other rings and fields. For instance, we now have sum-product estimates for complex numbers (see [3,7,9]), quaternions (see [1,3,12]), square matrices (see [4,[10][11][12][13]) and Function fields (see [2]).…”

Growth in Sumsets of Higher Convex Functions

“…The sum-product problem has been studied in other fields as well. We particularly note that for A ⊂ C, (3) is known for all δ < 1/3 + c (for some small c) [2] and for subsets of a function field A ⊂ F q ((t −1 )), it is known for all δ < 1/5 (with the implied constant C also depending on q) [3]. In this paper, we prove a related sum-product type result in each setting.…”

Sum-Product Estimates for Diagonal Matrices

“…In recent years, considerable work has also been done to extend these bounds to other rings and fields. For instance, there are sum-product estimates for complex numbers (see [3,7,9]), quaternions (see [1,3,12]), square matrices (see [4,[10][11][12][13]) and function fields (see [2]).…”