Lower bounds for incidences with hypersurfaces

TL;DR

This paper develops a technique to establish lower bounds for incidences with various hypersurfaces in higher dimensions, demonstrating the tightness of known upper bounds and advancing understanding of incidence geometry.

Contribution

It introduces the first non-trivial lower bounds for incidence problems in ${\mathbb R}^d$, applicable to many hypersurfaces, and refines bounds related to the absence of complete bipartite subgraphs in incidence graphs.

Findings

Established lower bounds for incidences with hypersurfaces in ${\mathbb R}^d$

Showed some upper bounds are tight up to an epsilon in the exponent

Provided improved bounds for incidence graphs avoiding $K_{s,s}$

Abstract

We present a technique for deriving lower bounds for incidences with hypersurfaces in ${\mathbb R}^d$ with $d\ge 4$. 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 ${\mathbb R}^d$ are tight up to an extra $\varepsilon$ in the exponent. Specifically, for every $m$, $d\ge 4$, and $\varepsilon>0$ there exist $m$ points and $n$ hypersurfaces in ${\mathbb R}^d$ (where $n$ depends on $m$) with no $K_{2,\frac{d-1}{\varepsilon}}$ in the incidence graph and $\Omega\left(m^{(2d-2)/(2d-1)}n^{d/(2d-1)-\varepsilon} \right)$ incidences. Moreover, we provide improved lower bounds for the case of no $K_{s,s}$ 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.