Abstract

We prove that if E⊂Fdq, d≥2, and F⊂Graff(d−1,d), the set of affine d−1-dimensional planes in Fdq, then |Δ(E,F)|≥q/2 if |E||F|>qd+1, where Δ(E,F) is the set of distances from points in E to planes in F. In dimension three and higher this significantly improves the exponent obtained by Pham, Phuong, Sang, Valculescu and Vinh (2018), who obtain the same conclusion under the assumption |E||F|≥Cq4d/3.