An improved sum-product bound for quaternions

TL;DR

This paper establishes a stronger sum-product inequality for finite sets of quaternions, extending previous results from real numbers to quaternions, with a new constant improving the bound.

Contribution

It introduces an improved sum-product bound for quaternions, generalizing earlier real number results and demonstrating a new lower bound with an absolute constant.

Findings

Sum of set and its additive inverse is large

Product set of the set is large

Combined sum-product bound exceeds previous limits

Abstract

We show that there exists an absolute constant $c > 0$, such that, for any finite set $A$ of quaternions, \[ \max\{|A+A, |AA| \} \gtrsim |A|^{4/3 + c}. \] This generalizes a sum-product bound for real numbers proved by Konyagin and Shkredov.