Exploration on Incidence Geometry and Sum-Product Phenomena

Here we give a short survey of our new results. References to the complete proofs can be found in the text of this article and in the litterature.

In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log |A|}$. The latter inequality had in fact already been established in an earlier work of the author and Rudnev (arXiv:1203.6237), which built upon the recent developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s dist...

For sufficiently large $n$, we show that in every configuration of $n$ points chosen inside the unit square there exists a triangle of area less than $n^{-8/7-1/2000}$. This improves upon a result of Koml\'os, Pintz and Szemer\'edi from 1982. Our approach establishes new connections between the Heilbronn triangle problem and various themes in incidence geometry and projection theory which are closely related to the discretized sum-product phenomenon.

We prove a new Elekes-Szab\'o type estimate on the size of the intersection of a Cartesian product $A\times B\times C$ with an algebraic surface $\{f=0\}$ over the reals. In particular, if $A,B,C$ are sets of $N$ real numbers and $f$ is a trivariate polynomial, then either $f$ has a special form that encodes additive group structure (for example $f(x,y,x) = x + y - z$), or $A \times B\times C \cap\{f=0\}$ has cardinality $O(N^{12/7})$. This is an improvement over the previous...

This is a survey of open problems in different parts of combinatorial and additive number theory. The paper is based on lectures at the Centre de Recerca Matematica in Barcelona on January 23 and January 25, 2008.

We estimate the number of incidences in a configuration of $m$ lines and $n$ points in dimension 3. The main term is $mn^{1/3}$ if we work over the real or complex numbers but $mn^{2/5}$ over finite fields. Both of these are optimal, aside from a multiplicative constant that is at most 5.

Let $I(n,l)$ denote the maximum possible number of incidences between $n$ points and $l$ lines. It is well known that $I(n,l) = \Theta(n^{2/3}l^{2/3} + n + l)$. Let $c_{\mathrm{SzTr}}$ denote the lower bound on the constant of proportionality of the $n^{2/3}l^{2/3}$ term. The known lower bound, due to Elekes, is $c_{\mathrm{SzTr}} \ge 2^{-2/3} = 0.63$. With a slight modification of Elekes' construction, we show that it can give a better lower bound of $c_{\mathrm{SzTr}} \ge 1...

Let $\mathbb{F}$ be a field, let $P \subseteq \mathbb{F}^d$ be a finite set of points, and let $\alpha,\beta \in \mathbb{F} \setminus \{0\}$. We study the quantity \[|\Pi_{\alpha, \beta}| = \{(p,q,r) \in P \times P \times P \mid p \cdot q = \alpha, p \cdot r = \beta \}.\] We observe a connection between the question of placing an upper bound on $|\Pi_{\alpha,\beta}|$ and a well-studied question on the number of incidences betwen points and hyperplanes, and use this connecti...

We improve the well-known Szemer\'edi-Trotter incidence bound for proper 3--dimensional point sets (defined appropriately)

Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements. We prove an identity for the second moment of its incidence function and deduce a variety of existing results from the literature, not all naturally associated with lines in $\mathbb{F}_q^2$, in a unified and elementary way.