On the two-parameter erdos-falconer distance problem in finite fields

Given E subset of F-q(d) x F-q(d), with the finite field F-q of order q and the integer d >= 2, we define the two-parameter distance set Delta(d,d)(E) = {(parallel to x - y parallel to, parallel to z - t parallel to) : (x,z), (y, t) is an element of E}. Birklbauer and Iosevich ['A two-parameter finite field Erdos-Falconer distance problem', Bull. Hellenic Math. Soc. 61 (2017), 21-30] proved that if vertical bar E vertical bar >> q((3d+1)/2), then vertical bar Delta(d,d)(E)vertical bar = q(2). For d = 2, they showed that if vertical bar E vertical bar >> q(10/3), then vertical bar Delta(2,2)(E)vertical bar >> q(2). In this paper, we give extensions and improvements of these results. Given the diagonal polynomial P(x) = Sigma(d)(i=1) a(i)x(i)(s) is an element of F-q[X-1, ..., x(d)], the distance induced by P over F-q(d) is parallel to x - y parallel to(s) := P(x - y), with the corresponding distance set Delta(s)(d,d)(E) = {(parallel to x - y parallel to(s), parallel to z - t parallel to(s)) : (x, z), (y, t) is an element of E}. We show that if vertical bar E vertical bar >> q((3d+1)/2), then vertical bar Delta(s)(d,d)(E)vertical bar >> q(2). For d = 2 and the Euclidean distance, we improve the former result over prime fields by showing that vertical bar Delta(2,2)(E)vertical bar >> p(2) for vertical bar E vertical bar >> p(13/4).