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) is an element of Z(p)[x], Gyarmati studied the solvability of the equations a + b = h(x), resp ab = h(x) with a is an element of A, b is an element of B, x is an element of Z(p), and for large subsets A, B, C, D of Z(p) Sirkozy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a is an element of A, b is an element of B, c is an element of C, d is an element of 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.