In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Zp and for h(x) ∈ Zp[x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with a ∈ A, b ∈ B, x ∈ Zp, and for large subsets A, B, C, D of Zp 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.