On The Solvability of Bilinear Equations in Finite Fields

We consider the equation $$ab + cd = \lambda, \qquad a\in A, b \in B, c\in C, d \in D,$$ over a finite field $F_q$ of $q$ elements, with variables from arbitrary sets $A, B, C, D \subseteq F_q$. The question of solvability of such and more general equations has recently been considered by D. Hart and A. Iosevich, who, in particular, proved that if #A #B #C #D \gg q^3, then above equation has a solution for any $\lambda \in F_q^*$. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.