Highly incidental patterns on a quadratic hypersurface in R4.

InźSharir and Solomon (2015), Sharir and Solomon showed that the number of incidences between m distinct points and n distinct lines in R 4 is (1) O ź m 2 ź 5 n 4 ź 5 + m 1 ź 2 n 1 ź 2 q 1 ź 4 + m 2 ź 3 n 1 ź 3 s 1 ź 3 + m + n , provided that no 2-flat contains more than s lines, and no hyperplane or quadric contains more than q lines, where the O ź hides a multiplicative factor of 2 c log m for some absolute constant c .In this paper we prove that, for integers m , n satisfying n 9 ź 8