On the solvability of Bilinear equations in finite fields

We consider the equation ab + cd = lambda, a is an element of A, b is an element of B, c is an element of C, d is an element of D, over a finite field F(q) of q elements, with variables from arbitrary sets A, B, C, D subset of F(q), The question of solvability of such and more general equations has recently been considered by Hart and Iosevich, who, in particular, prove that if #A#B#C#D >= Cq(3), for some absolute constant C > 0, then above equation has a solution for any lambda is an element of F(q)*. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.