Conditional expanding bounds for two-variables functions over prime fields

Abstract: In this paper we provide in $\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in $\bFp^*$ is the existenceness of $\Delta(\alpha)>0$ such that if $|A|\asymp p^{\alpha}$ then $$ \max(|A+A|,|A\cdot A|)\gg |A|^{1+\Delta(\alpha)}, $$ Our aim is to obtain analogous results for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$: if $|A|\asymp|B|\asymp p^{\alpha}$ then $$ \max(|f(A,B)|,|g(A,B)|)\gg |A|^{1+\Delta(\alpha)} $$ for some $\Delta(\alpha)>0$.