arXiv subjects

Audie Warren

Publications and source records attributed to Audie Warren.

25 records · Page 2Linked to original sources

arXiv2020

An Energy Bound in the Affine Group

We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set P for which no line contains a positive proportion of points from P, there may be at most one line, meeting the set of lines defined by P in at most a constant multiple of |P| points.

math.CO↗

arXiv2020

The Spherical Kakeya Problem in Finite Fields

We study subsets of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$, for odd $q$, which contain either a sphere for each radius or a sphere for each first coordinate of the center. We call such sets radii spherical Kakeya sets and center spherical Kakeya sets, respectively. For $n\ge 4$ we prove a general lower bound on the size of any set containing $q-1$ different spheres which applies to both kinds of spherical Kakeya sets. We provide constructions which meet the main terms of this lower bound. We also give a construction showing that we cannot get a lower bound of order of magnitude~$q^n$ if we take lower dimensional objects such as circles in $\mathbb{F}_q^3$ instead of spheres, showing that there are significant differences to the line Kakeya problem. Finally, we study the case of dimension $n=1$ which is different and equivalent to the study of sum and difference sets that cover $\mathbb{F}_q$.

math.CO↗

arXiv2019

Conical Kakeya and Nikodym Sets in Finite Fields

A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.) Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.

math.CO↗

arXiv2019

New Expander Bounds from Affine Group Energy

The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of Rudnev and Shkredov. We bound the energy of several systems of lines, in some cases obtaining a better energy bound than the corresponding bounds obtained by Rudnev and Shkredov by exploiting a connection with collinear quadruples. Our motivation for seeking to generalise and improve the incidence bound obtained by Rudnev and Shkredov comes from possible applications to sum-product problems. For example, we prove that, for any finite $A \subset \mathbb R$ the following superquadratic bound holds: \[ \left| \left \{ \frac{ab-cd}{a-c} : a,b,c,d \in A \right \} \right| \gg |A|^{2+\frac{1}{14}}. \] This improves the previously known bound with exponent $2$. We also give a threshold-beating asymmetric sum-product estimate for sets with small sum set by proving that there exists a positive constant $c$ such that for all finite $A,B \subset \mathbb R$, \[ |A+A| \ll K|A| \Rightarrow |AB| \gg_K |A||B|^{1/2+c}. \]

math.CO↗

arXiv2019

On Products of Shifts in Arbitrary Fields

We adapt the approach of Rudnev, Shakan, and Shkredov to prove that in an arbitrary field $\mathbb{F}$, for all $A \subset \mathbb{F}$ finite with $|A| < p^{1/4}$ if $p:= Char(\mathbb{F})$ is positive, we have $$|A(A+1)| \gtrsim |A|^{11/9}, \qquad |AA| + |(A+1)(A+1)| \gtrsim |A|^{11/9}.$$ This improves upon the exponent of $6/5$ given by an incidence theorem of Stevens and de Zeeuw.

math.CO↗

arXiv2018

Constructions for the Elekes-Szabó and Elekes-Rónyai problems

We give a construction of a non-degenerate polynomial $F\in \mathbb R[x,y,z]$ and a set $A$ of cardinality $n$ such that $\left|Z(F)\cap (A \times A \times A) \right| \gg n^{\frac{3}{2}}$, thus providing a new lower bound construction for the Elekes--Szabó problem. We also give a related construction for the Elekes--Rónyai problem restricted to a subgraph. This consists of a polynomial $f\in \mathbb R[x,y]$ that is not additive or multiplicative, a set $A$ of size $n$, and a subset $P\subset A\times A$ of size $|P|\gg n^{3/2}$ on which $f$ takes only $n$ distinct values.

math.CO↗

arXiv2018

Improved Bounds for Pencils of Lines

We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $\mathbb R^2$, with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main result states that the number of such points is $O(n^{11/6})$, improving a result of Chang and Solymosi. We also consider constructions for this problem. Alon, Ruzsa and Solymosi constructed an arrangement of four non-collinear $n$-pencils which determine $Ω(n^{3/2})$ four-rich points. We give a construction to show that this is not tight, improving this lower bound by a logarithmic factor. We also give a construction of a set of $m$ $n$-pencils, whose centres are in general position, that determine $Ω_m(n^{3/2})$ $m$-rich points.

math.CO↗