An Improved Point-Line Incidence Bound Over Arbitrary Fields

We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most $O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive characteristic $p$, we assume $m^{-2}n^{13}\ll p^{15}$). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.Comment: 18 pages. To appear in the Bulletin of the London Mathematical Societ

Extensions of a result of Elekes and R\'onyai

Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and R\'onyai proved that if the graph of a polynomial contains $cn^2$ points of an $n\times n\times n$ cartesian product in $\mathbb{R}^3$, then the polynomial has the form $f(x,y)=g(k(x)+l(y))$ or $f(x,y)=g(k(x)l(y))$. They used this to prove a conjecture of Purdy which states that given two lines in $\mathbb{R}^2$ and $n$ points on each line, if the number of distinct distances between pairs of points, one on each line, is at most $cn$, then the lines are parallel or orthogonal. We extend the Elekes-R\'onyai Theorem to a less symmetric cartesian product. We also extend the Elekes-R\'onyai Theorem to one dimension higher on an $n\times n\times n\times n$ cartesian product and an asymmetric cartesian product. We give a proof of a variation of Purdy's conjecture with fewer points on one of the lines. We finish with a lower bound for our main result in one dimension higher with asymmetric cartesian product, showing that it is near-optimal.Comment: 23 page

Bisector energy and few distinct distances

We introduce the bisector energy of an $n$-point set $P$ in $\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\frac{2}{5}}n^{\frac{12}{5}+\epsilon} + M(n)n^2).$. We also prove the lower bound $\Omega(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If $P$ determines $O(n/\sqrt{\log n})$ distinct distances, then for any $0<\alpha\le 1/4$, either there exists a line or circle that contains $n^\alpha$ points of $P$, or there exist $\Omega(n^{8/5-12\alpha/5-\epsilon})$ distinct lines that contain $\Omega(\sqrt{\log n})$ points of $P$. This result provides new information on a conjecture of Erd\H{o}s regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains $M(n)$ points of $P$, then the number of distinct perpendicular bisectors determined by $P$ is $\Omega(\min\{M(n)^{-2/5}n^{8/5-\epsilon}, M(n)^{-1} n^2\})$. This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over $\mathbb{R}$, initiated by Elekes and R\'onyai.Comment: 18 pages, 2 figure

On the number of ordinary conics

We prove a lower bound on the number of ordinary conics determined by a finite point set in $\mathbb{R}^2$. An ordinary conic for a subset $S$ of $\mathbb{R}^2$ is a conic that is determined by five points of $S$, and contains no other points of $S$. Wiseman and Wilson proved the Sylvester-Gallai-type statement that if a finite point set is not contained in a conic, then it determines at least one ordinary conic. We give a simpler proof of their result and then combine it with a result of Green and Tao to prove our main result: If $S$ is not contained in a conic and has at most $c|S|$ points on a line, then $S$ determines $\Omega_c(|S|^4)$ ordinary conics. We also give a construction, based on the group structure of elliptic curves, that shows that the exponent in our bound is best possible