2018/06/09 by Rudnev, Misha
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1806.03534
The point-plane incidence theorem states that the number of incidences between n points and m≥ n planes in the projective three-space over a field F, is O(m√(n)+ m k), where k is the maximum number of collinear points, with the extra condition n< p2 if F has characteristic p>0. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in F2, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in Fd, for d=2,3,4. These are the problem of the minimum number of distinct nonzero values of a non-degenerate bilinear form on a point set in d=2, the analogue of the Erd\H os distinct distance problem in d=2,3 and additive energy estimates for sets, supported on a paraboloid and sphere in d=3,4. It avoids discussing sum-product type problems (corresponding to the special case of incidences with Cartesian products), which have lately received more attention.