One Arithmetic Gadget, Two Problems

There is an absolute constant $c>0$ such that, for every $0<\epsilon \le 1$, arbitrarily large finite sets $A\subset \mathbb{C}$ satisfy $$\mathrm{\mid }A+A\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^{1+\epsilon },\mathrm{\mid }AA\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^{2-c\epsilon },{\nu }_1(A)\ge \mathrm{\mid }A{\mathrm{\mid }}^{1+c\epsilon },$$ where ${\nu }_1(A)$ counts unordered unit-distance pairs. We combine the arithmetic directions from the recent unit-distance construction with the multiplicative enlargement used in the recent sum-product construction. The arithmetic ingredients are stated as explicit inputs.