On character sums with distances on the upper half plane over a finite field

For the finite field Fq of q elements (q odd) and a quadratic non-residue (that is, a non-square) α∈Fq, we define the distance functionδ(u+vα,x+yα)=(u−x)2−α(v−y)2vy on the upper half plane Hq={x+yα|x∈Fq,y∈Fq∗}⊆Fq2. For two sets E,F⊂Hq with #E=E, #F=F and a non-trivial additive character ψ on Fq, we give the following estimate|∑w∈E,z∈Fψ(δ(w,z))|⩽min{q+2qE,3q5/4}EF, which (assuming that F⩾E) is non-trivial if EF/q2→∞ as q→∞.