2020/03/10 by Rudnev, Misha
#11B75 #68R05 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2003.04744
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(n1/2m3/4 + m+n) for the number of incidences between n lines in a complex and m points in \mathbb F3, where \mathbb F is a field, and n≤ char(\mathbb 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 \mathbb F3. This implied the new bound O(n3/2) for the number of realisations of the distance, which is a square, for \mathbb F, where -1 is not a square in the \mathbb F-analogue of the Erd\H os single distance problem in \mathbb R3. Our incidence bound yields, under a natural constraint, a weaker bound O(n1.6), which holds for any distance, including zero, over any \mathbb F.