Lower bounds for incidences with hypersurfaces

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

Higher Distance Energies and Expanders with Structure

We adapt the idea of higher moment energies, originally used in Additive Combinatorics, so that it would apply to problems in Discrete Geometry. This new approach leads to a variety of new results, such as (i) Improved bounds for the problem of distinct distances with local properties. (ii) Improved bounds for problems involving expanding polynomials in ${\mathbb R}[x,y]$ (Elekes-Ronyai type bounds) when one or two of the sets have structure. Higher moment energies seem to be related to additional problems in Discrete Geometry, to lead to new elegant theory, and to raise new questions

Fast domino tileability

Domino tileability is a classical problem in Discrete Geometry, famously solved by Thurston for simply connected regions in nearly linear time in the area. In this paper, we improve upon Thurston's height function approach to a nearly linear time in the perimeter.Comment: Appeared in Discrete Comput. Geom. 56 (2016), 377-39

Point-curve incidences in the complex plane

We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big)$. We establish the slightly weaker bound $O_\varepsilon\big(m^{\frac{k}{2k-1}+\varepsilon}n^{\frac{2k-2}{2k-1}}+m+n\big)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in ${\mathbb C}^2$ with $k$ degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over ${\mathbb C}$.Comment: The proof was significantly simplified, and now relies on the Picard-Lindelof theorem, rather than on foliation