One Arithmetic Gadget, Two Problems

Abstract: There is an absolute constant $c>0$ such that, for every $0<\varepsilon\leq1$, arbitrarily large finite sets $A\subset\mathbb{C}$ satisfy [ |A+A|\leq |A|^{1+\varepsilon},\qquad |AA|\leq |A|^{2-c\varepsilon},\qquad ν_1(A)\geq |A|^{1+c\varepsilon}, ] where $ν_1(A)$ 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.