11. Point-plane incidences and some applications in positive characteristic

Abstract

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 , iswhere k is the maximum number of collinear points, with the extra condition n < p 2 if F has characteristic p > 0. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in F 2 , 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 F d , 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 Erdős 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.2000 Mathematics Subject Classification. 68R05,11B75.1 the folklore that the proof should work, with some constraints, over a general field. The first "official" account of this was given by Ellenberg and Hablicsek [16] in late 2013, followed by Kollár [28] and the author [39] in 2014. The latter two had been aware of a 2003 paper by Voloch [50], which discussed the constraints under which the key element of the First Guth-Katz theorem proof, the Monge-Salmon theorem [42], applied in positive characteristic.Theorem 2 (First Guth-Katz theorem). Let L be a set of lines in C 3 . Suppose, no more then two lines are concurrent. Then the number of pair-wise intersections of lines in L is O |L| 3 2 + |L|k ,

On Restriction Estimates for the Zero Radius Sphere over Finite Fields

“…The key new ingredient is improved additive energy estimates for subsets of spheres in even dimensions. In particular, our additive energy estimate improves and extends Rudnev's recent work [15] in four dimensions to higher even dimensions.As the most interesting result of this paper, we prove that if −1 is not a square number of F * q and the dimension d is 4k + 2 for some k ∈ N, then the L 2 → L (2d+4)/d extension estimate for spheres of zero radius holds. This result is also sharp and provides us of an interesting fact that the L 2 → L r extension estimate for zero spheres with specific assumptions is much better than the Stein-Tomas result which can not be improved in general for cones or zero spheres in even dimensions.2010 Mathematics Subject Classification.…”

“…The key new ingredient is improved additive energy estimates for subsets of spheres in even dimensions. In particular, our additive energy estimate improves and extends Rudnev's recent work [15] in four dimensions to higher even dimensions.…”

“…It is observed in [15] that if x, y, z ∈ A ⊂ S j with x + y − z ∈ S j , then (x − z) · (y − z) = 0. Indeed, suppose that x, y, z ∈ A ⊂ S j and x + y − z ∈ S j .…”

“…Remark 2.6. Rudnev [15] recently obtained additive energy estimates for subsets of spheres with non-zero radius in three and four dimensions. In particular, in four dimensions, he proved the following result.…”

Extension theorems in vector spaces over finite fields

“…Let g be a primitive element of F q , and S g the sphere of radius g centered at the origin in F 3 q . For A ⊂ S g of size n, we have This lemma can be proved by using the energy for small sets in [49,Theorem 18]. Hence, we leave the details to readers.…”