Equations in finite fields with restricted solution sets. I (Character sums)

In earlier papers, for “large” (but otherwise unspecified) subsets A , B of Z p and for h ( x ) ∈ Z p [ x ], Gyarmati studied the solvability of the equations a + b = h ( x ), resp. ab = h ( x ) with a ∈ A , b ∈ B , x ∈ Z p , and for large subsets A , B , C , D of Z p Sárközy showed the solvability of the equations a + b = cd , resp. ab + 1 = cd with a ∈ A , b ∈ B , c ∈ C , d ∈ D . In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.