Conditional expanding bounds for two-variable functions over prime fields

In this paper we provide in F p 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 immensely studied in the recent past. A typical result in F p ∗ is the existence of Δ ( α ) > 0 such that if | A | ≍ p α then max ( | A + A | , | A ⋅ A | ) ≫ | A | 1 + Δ ( α ) , Our aim is to obtain analogous results for related pairs of two-variable functions f ( x , y ) and g ( x , y ) : if | A | ≍ | B | ≍ p α then max ( | f ( A , B ) | , | g ( A , B ) | ) ≫ | A | 1 + Δ ( α ) for some Δ ( α ) > 0 .