Sum-product phenomenon

The sum-product phenomenon says that a finite subset of a field cannot usually have both a small sumset and a small product set unless it has additional subfield-like structure.

For finite $U,V,W\subseteq \mathbb{C}$ with $U\ne \{0\}$ and $W\ne \{0\}$,In particular, addition and multiplication cannot both expand a nontrivial finite set of complex numbers only slightly.