  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.

Combinatorics