Counting joints in vector spaces over arbitrary fields

We give a proof of the "folklore" theorem that the Kaplan--Sharir--Shustin/Quilodr\'an result on counting joints associated to a family of lines holds in vector spaces over arbitrary fields, not just the reals. We also discuss a distributional estimate on the multiplicities of the joints in the case that the family of lines is sufficiently generic.Comment: Not intended for publication. References added and other minor edits in this versio

Improved rank bounds for design matrices and a new proof of Kelly's theorem

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in which they were used to answer questions regarding point configurations. In this work we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai theorem

Variations on the Sum-Product Problem

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\subset{\mathbb{R}}$, $$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$$ giving a partial answer to a conjecture of Balog. In a similar spirit, it is established that $$|A(A+A+A+A)|\gg{\frac{|A|^2}{\log{|A|}}},$$ a bound which is optimal up to constant and logarithmic factors. We also prove several new results concerning sum-product estimates and expanders, for example, showing that $$|A(A+a)|\gg{|A|^{3/2}}$$ holds for a typical element of $A$.Comment: 30 pages, new version contains improved exponent in main theorem due to suggestion of M. Z. Garae

Sharp Szemer\'{e}di-Trotter constructions from arbitrary number fields

In this note, we describe an infinite family of sharp Szemer\'{e}di-Trotter constructions. These constructions are cartesian products of arbitrarily high dimensional generalized arithmetic progressions (GAPs), where the bases for these GAPs come from arbitrary number fields over $\mathbb{Q}$. This can be seen as an extension of a recent result of Guth and Silier, who provided similar constructions based on the field $\mathbb{Q}(\sqrt{k})$ for square-free $k$. However, our argument borrows from an idea of Elekes which produces unbalanced grids, instead of the balanced ones given by Guth and Silier. This simplifies the analysis and allows us to easily give constructions coming from any number field.Comment: 4 pages, comments welcome

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