2015/11/10 by Adam Sheffer, Sheffer, Adam · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1511.03298
arxiv created 2016/10/04 · arxiv updated 2016/10/05
We present a technique for deriving lower bounds for incidences with hypersurfaces in \mathbb Rd with d≥ 4. These bounds apply to a large variety of hypersurfaces, such as hyperplanes, hyperspheres, paraboloids, and hypersurfaces of any degree. Beyond being the first non-trivial lower bounds for various incidence problems, our bounds show that some of the known upper bounds for incidence problems in \mathbb Rd are tight up to an extra ε in the exponent. Specifically, for every m, d≥ 4, and ε>0 there exist m points and n hypersurfaces in \mathbb Rd (where n depends on m) with no K2,(d-1)/(ε) in the incidence graph and Ω(m(2d-2)/(2d-1)nd/(2d-1)-ε ) incidences. Moreover, we provide improved lower bounds for the case of no Ks,s in the incidence graph, for large constants s. Our analysis builds upon ideas from a recent work of Bourgain and Demeter on discrete Fourier restriction to the four- and five-dimensional spheres. Specifically, it is based on studying the additive energy of the integer points in a truncated paraboloid.