On a certain sum-product estimate in fields of prime order

Abstract

Let \(\mathbb{F}_p \) be the finite field of p elements with p prime. If A is a subset of \(\mathbb{F}_p \) and g is an element of \(\mathbb{F}_p^* \) with order ν, then $$\max \{ |A + g \cdot A|,|A \cdot A|\} \gg \left( {\frac{\nu } {{\nu + |A|^2 }}} \right)^{1/12} |{\rm A}|^{13/12} $$ .