Optimal simultaneous tradeoff between sumset, product-set, and unit-distance gains

Determine the best simultaneous tradeoff between the product-set saving and unit-distance gain for finite sets of complex numbers subject to the constraint |A+A|\leq |A|^{1+\varepsilon}.

Background

The main theorem constructs arbitrarily large finite sets of complex numbers A satisfying |A+A|\leq |A|^{1+\varepsilon}, while simultaneously obtaining a product-set bound |AA|\leq |A|^{2-c\varepsilon} and a unit-distance bound \nu_1(A)\geq |A|^{1+c\varepsilon}, for an absolute constant c>0. The authors explicitly leave unresolved the problem of determining the optimal relationship between the exponents governing the product-set saving and the unit-distance excess under the prescribed small-sumset condition. They note that the argument establishes only positive constants and does not optimize these gains.