A sum-product estimate in finite fields, and applications

Jean Bourgain, Nets Katz, Terence Tao

Let $A$ be a subset of a finite field $F:=\mathbb{Z}/q\mathbb{Z}$ for some prime $q$. If $|F{|}^{\delta }<|A|<|F{|}^{1-\delta }$ for some $\delta >0$, then we prove the estimate $|A+A| + |A.A| \geq c(δ) |A|^{1+\eps…