A Szemeredi-Trotter type theorem in R4\mathbb{R}^4

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemer\'edi-Trotter type theorem for 2--planes in $\mathbb{R}^4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemer\'edi-Trotter type theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by T\'oth using a different method). As a third special case, we obtain a Szemer\'edi-Trotter type theorem for complex unit circles in $\mathbb{C}^2$. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.Comment: 50 pages. V3: final version. To appear in Discrete and Computational Geometr

An improved bound on the number of point-surface incidences in three dimensions

We show that $m$ points and $n$ smooth algebraic surfaces of bounded degree in $\mathbb{R}^3$ satisfying suitable nondegeneracy conditions can have at most $O(m^{\frac{2k}{3k-1}}n^{\frac{3k-3}{3k-1}}+m+n)$ incidences, provided that any collection of $k$ points have at most O(1) surfaces passing through all of them, for some $k\geq 3$. In the case where the surfaces are spheres and no three spheres meet in a common circle, this implies there are $O((mn)^{3/4} + m +n)$ point-sphere incidences. This is a slight improvement over the previous bound of $O((mn)^{3/4} \beta(m,n)+ m +n)$ for $\beta(m,n)$ an (explicit) very slowly growing function. We obtain this bound by using the discrete polynomial ham sandwich theorem to cut $\mathbb{R}^3$ into open cells adapted to the set of points, and within each cell of the decomposition we apply a Turan-type theorem to obtain crude control on the number of point-surface incidences. We then perform a second polynomial ham sandwich decomposition on the irreducible components of the variety defined by the first decomposition. As an application, we obtain a new bound on the maximum number of unit distances amongst $m$ points in $\mathbb{R}^3$.Comment: 17 pages, revised based on referee comment

On the number of rich lines in truly high dimensional sets

We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\ldots,v_n \in \mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemer\'edi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. This conjecture was shown to hold in $\mathbb{R}^3$ in the seminal work of Guth and Katz \cite{GK10} and was also recently proved over $\mathbb{R}^4$ (under some additional restrictions) \cite{SS14}. For the special case of arithmetic progressions ($r$ collinear points that are evenly distanced) we give a bound that is tight up to low order terms, showing that a $d$-dimensional grid achieves the largest number of $r$-term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree $r-2$ Veronese embedding takes $r$-collinear points to $r$ linearly dependent images. Hence, each collinear $r$-tuple of points, gives us a dependent $r$-tuple of images. We then use the design-matrix method of \cite{BDWY12} to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set