An improved sum-product bound for quaternions

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theorem_1_2: States that there is an absolute constant c > 0 such that every finite set A of quaternions satisfies |A + A| + |AA| >> |A|^(4/3 + c), with c not made explicit.

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Abdul Basit, Ben Lund, "An improved sum-product bound for quaternions," SIAM J. Discrete Math. 33 (2019), no. 2, 1044--1060, DOI 10.1137/18M1231468.

The copy read for this card is the arXiv preprint, arXiv:1809.02214v2 (10 November 2021).

Theorem 1.2 (p. 2) states that there is an absolute constant c > 0 such that every finite set A of quaternions satisfies |A + A| + |AA| >> |A|^(4/3 + c); the abstract (p. 1) states it as max{|A + A|, |AA|} >~ |A|^(4/3 + c), a form that allows a logarithmic loss. This passes the exponent 4/3, which Solymosi and Wong reached for quaternions up to an epsilon loss. The proof follows Konyagin and Shkredov's real-number argument and splits on the additive energy of A: when the energy is small a Solymosi-type geometric argument in the spirit of Konyagin–Rudnev and Solymosi–Wong applies (Section 4.3, pp. 13--17), and when it is large Konyagin and Shkredov's additive-combinatorial argument is adapted to a noncommutative ring (Section 4.2, pp. 12--13), with the energy bounds of Section 3 (pp. 4--10) replacing the Szemerédi–Trotter theorem by a Solymosi–Tao incidence bound and reworking the relevant definitions for noncommutative multiplication. The authors make no attempt to find the largest c and give no value for it. For real numbers the paper cites Shakan's exponent 4/3 + 5/5277 as the best known (p. 2).

Read status: claims checked for Theorem 1.2, its statement read clause by clause on p. 2 of arXiv v2; the proof was read for its structure only.

Source: https://arxiv.org/abs/1809.02214. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1809.02214), every other right reserved.

Results.

• Theorem 1.2 (p. 2): there is an absolute constant c > 0 such that every finite set A of quaternions has |A + A| + |AA| >> |A|^(4/3 + c).

Bears on.

• Problem 52: since the integers lie in the quaternions, Theorem 1.2 gives max(|A + A|, |AA|) >> |A|^(4/3 + c) for every finite set A of integers, with c > 0 absolute but unspecified; the problem asks for the exponent 2 - eps for every eps > 0, which the theorem does not decide.

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