9 citations · 29 across the 24 of their papers we have counts for
20 papers · 1 filter
Multi-parameter Szemerédi-Trotter-type theorems and applications in finite fields
Hung Le, Steven Senger, Minh-Quan Vo
We prove some novel multi-parameter point-line incidence estimates in vector spaces over finite fields. While these could be seen as special cases of higher-dimensional incidence r…
VC-dimension and pseudo-random graphs
Thang Pham, Steven Senger, Michael Tait +1
Let be a graph and be a set of vertices. For each , let be the function defined by \[h_v(u)=\begin{cases} &1 ~\mbox{if}~u\sim…
An improved point-line incidence bound over arbitrary finite fields via the VC-dimension theory
Alex Iosevich, Thang Pham, Steven Senger +1
The main purpose of this paper is to prove that the point-line incidence bound due to Vinh (2011) over arbitrary finite fields can be improved in certain ranges by using tools from…
Geometric structures in pseudo-random graphs
Thang Pham, Steven Senger, Michael Tait +1
In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the fi…
The Erdos-Falconer distance problem in the tree setting
Thang Pham, Steven Senger, Dung The Tran
The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set , if the Hausdorff dimension of…
Convexity, Superquadratic Growth, and Dot Products
Brandon Hanson, Oliver Roche-Newton, Steven Senger
Let be a point set with cardinality . We give an improved bound for the number of dot products determined by , proving that, \[ |\{ p \cdot q :p,q \in…
Angle chains and pinned variants
Eyvindur Ari Palsson, Steven Senger, Charles Wolf
We study a variant of the Erd\H os unit distance problem, concerning angles between successive triples of points chosen from a large finite point set. Specifically, given a large f…
On the number of dot product chains in finite fields and rings
Vincent Blevins, David Crosby, Ethan Lynch +1
We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either or $\m…
Bounds on point configurations determined by distances and dot products
Slade Gunter, Eyvi Palsson, Ben Rhodes +1
We study a family of variants of Erd\H os' unit distance problem, concerning distances and dot products between pairs of points chosen from a large finite point set. Specifically,…
Dot product chains
Shelby Kilmer, Caleb Marshall, Steven Senger
We study a variant of Erd\H os' unit distance problem, concerning dot products between successive pairs of points chosen from a large finite point set. Specifically, given a large…
Uniqueness of Optimal Point Sets Determining Two Distinct Triangles
Hazel N. Brenner, James S. Depret-Guillaume, Eyvindur A. Palsson +1
In this paper, we show that the maximum number of points in dimensions determining exactly 2 distinct triangles is . We further show that this maximum is uniquely achi…
On the Number of Discrete Chains
Eyvindur Ari Palsson, Steven Senger, Adam Sheffer
We study a generalization of Erd\H os's unit distances problem to chains of distances. Given a set of points, and a sequence of distances ,…
Pairs of dot products in finite fields and rings
David Covert, Steven Senger
We obtain bounds on the number of triples that determine a given pair of dot products arising in a vector space over a finite field or a module over the set of integers modulo a po…
Upper bounds on pairs of dot products
Daniel Barker, Steven Senger
Given a large finite point set, , we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair…
A note on the multiplicative structure of an additively shifted product set, AA+1
Steven Senger
We consider the multiplicative structure of sets of the form AA+1, where where A is a large, finite set of real numbers. In particular, we show that the additively shifted product…
On the number of distinct values of a class of functions with finite domain
Robert Coulter, Steven Senger
By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are e…
A Furstenberg-Katznelson-Weiss type theorem on (d + 1)-point configurations in sets of positive density in finite field geometries
David Covert, Derrick Hart, Alex Iosevich +2
We show that if , the -dimensional vector space over the finite field with elements, and , where , the…
Sharpness of Falconer's estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains
Alex Iosevich, Steven Senger
In this paper we prove, for all , that for no does imply the canonical Falconer distance problem incidence bound, or the analogous estima…
Orthogonal systems in vector spaces over finite fields
Alex Iosevich, Steve Senger
We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors.
Distance graphs in vector spaces over finite fields, coloring and pseudo-randomness
Derrick Hart, Alex Iosevich, Doowon Koh +2
In this paper we systematically study various properties of the distance graph in , the -dimensional vector space over the finite field with eleme…