We prove a new upper bound on the number of @math -rich lines (lines with at least @math points) in a truly' @math -dimensional configuration of points @math . More formally, we show that, if the number of @math -rich lines is significantly larger than @math then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor @math can be replaced with a tight @math . If true, this would generalize the classic Szemer 'edi-Trotter theorem which gives a bound of @math on the number of @math -rich lines in a planar configuration. This conjecture was shown to hold in @math in the seminal work of Guth and Katz GK10 and was also recently proved over @math (under some additional restrictions) SS14 . For the special case of arithmetic progressions ( @math collinear points that are evenly distanced) we give a bound that is tight up to low order terms, showing that a @math -dimensional grid achieves the largest number of @math -term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree @math Veronese embedding takes @math -collinear points to @math linearly dependent images. Hence, each collinear @math -tuple of points, gives us a dependent @math -tuple of images. We then use the design-matrix method of BDWY12 to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.
Similarly, using the results of in @cite_10 , we can prove the following theorem from which a slightly weaker version of Conjecture in @math trivially follows (see Appendix ). We are not aware of any examples where points arranged on a quadric hypersurface in @math result in significantly more rich lines than in a four dimensional grid. It is, however, possible that one needs to weaken Conjecture so that for some @math , an @math -dimensional hypersurface of constant degree (possibly depending on @math ) contains @math points.
@cite_10
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We prove an incidence theorem for points and planes in the projective space @math over any field @math , whose characteristic @math An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving @math if @math . This yields a bound on the number of incidences between @math points and @math planes in @math , with @math as @math where @math is the maximum number of collinear planes, provided that @math if @math . Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when @math . For a non-collinear point set @math and a non-degenerate symmetric or skew-symmetric bilinear form @math , the number of distinct values of @math on pairs of points of @math is @math . This is also the best known bound over @math , where it follows from the Szemer 'edi-Trotter theorem. Also, a set @math , not supported in a single semi-isotropic plane contains a point, from which @math distinct distances to other points of @math are attained.