On incidences of lines in regular complexes. (December 2021) - Rudnev, Misha
Abstract: A regular linear line complex is a three-parameter set of lines in space, whose Plücker vectors lie in a hyperplane, which is not tangent to the Klein quadric. Our main result is a bound O ( n 1 / 2 m 3 / 4 + m + n ) for the number of incidences between n lines in a complex and m points in F 3, where F is a field, and n ≤ c h a r ( F ) 4 / 3 in positive characteristic. Zahl has recently observed that bichromatic pairwise incidences of lines coming from two distinct line complexes account for the nonzero single distance problem for a set of n points in F 3 . This implied the new bound O ( n 3 / 2 ) for the number of realisations of the distance, which is a square, for F, where − 1 is not a square in the F -analogue of the Erdős single distance problem in R 3 . Our incidence bound yields, under a natural constraint, a weaker bound O ( n 1 . 6 ), which holds for any distance, including zero, over any F .