An improved sum-product bound for quaternions

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