7 citations · 14 across the 11 of their papers we have counts for
18 papers · 1 filter
Point sets determining few angles are almost contained in a line or circle
Krishnendu Bhowmick, Oliver Roche-Newton, Audie Warren
We prove a structural theorem for point sets in which determine few pinned angles. More precisely, we prove the existence of an absolute constant such that if $…
On four-rich points defined by pencils
Michalis Kokkinos, Audie Warren
In this paper we study the number of four-rich points defined by pencils of certain algebraic objects. Our main result concerns the number of four-rich points defined by four sheav…
On Galois groups of type-1 minimally rigid graphs
Mehdi Makhul, Josef Schicho, Audie Warren
For every graph that is mimimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edg…
Incidences of Cubic Curves in Finite Fields
Audie Warren
In this paper we prove an incidence bound for points and cubic curves over prime fields. The methods generalise those used by Mohammadi, Pham, and Warren (2021).
A convex set with a rich difference
Oliver Roche-Newton, Audie Warren
We construct a convex set with cardinality and with the property that an element of the difference set can be represented in different ways. We also show that th…
A Point-Conic Incidence Bound and Applications over
Ali Mohammadi, Thang Pham, Audie Warren
In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on…
Incidences of Möbius transformations in
Audie Warren, James Wheeler
We develop the methods used by Rudnev and Wheeler to prove an incidence theorem between arbitrary sets of Möbius transformations and point sets in . We also note som…
Additive and multiplicative Sidon sets
Oliver Roche-Newton, Audie Warren
We give a construction of a set such that any subset with is neither an additive nor multiplicative Sidon set. In doing so…
On sum sets of convex functions
Sophie Stevens, Audie Warren
In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all finite sets, and for all convex or co…
The Elekes-Szabó Problem and the Uniformity Conjecture
Mehdi Makhul, Oliver Roche-Newton, Sophie Stevens +1
In this paper we give a conditional improvement to the Elekes-Szabó problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}…
The Spherical Kakeya Problem in Finite Fields
Mehdi Makhul, Audie Warren, Arne Winterhof
We study subsets of the -dimensional vector space over the finite field , for odd , which contain either a sphere for each radius or a sphere for each first coo…
Arcs in
Oliver Roche-Newton, Audie Warren
An arc is a subset of which does not contain any collinear triples. Let denote the number of arcs in with cardinality . This paper is pr…
An Energy Bound in the Affine Group
Giorgis Petridis, Oliver Roche-Newton, Misha Rudnev +1
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 a…
Conical Kakeya and Nikodym Sets in Finite Fields
Audie Warren, Arne Winterhof
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…
New Expander Bounds from Affine Group Energy
Oliver Roche-Newton, Audie Warren
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 o…
On Products of Shifts in Arbitrary Fields
Audie Warren
We adapt the approach of Rudnev, Shakan, and Shkredov to prove that in an arbitrary field , for all finite with if $p:= Char(\mat…
Constructions for the Elekes-Szabó and Elekes-Rónyai problems
Mehdi Makhul, Oliver Roche-Newton, Audie Warren +1
We give a construction of a non-degenerate polynomial and a set of cardinality such that $\left|Z(F)\cap (A \times A \times A) \right| \gg n^{\frac{…
Improved Bounds for Pencils of Lines
Oliver Roche-Newton, Audie Warren
We consider a question raised by Rudnev: given four pencils of concurrent lines in , with the four centres of the pencils non-collinear, what is the maximum possib…