arXiv subjects
Publications and source records attributed to Steven Senger.
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 the VC-dimension theory. As consequences, a number of applications will be discussed in detail.
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 finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.
The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set $Δ(A)$ has positive Lebesgue measure. In a very recent paper, Murphy, Petridis, Pham, Rudnev, and Stevens (2022) proved the prime field version of this result, namely, for $E\subset\mathbb{F}_p^2$ with $|E|\gg p^{5/4}$, there exist many points $x\in E$ such that the number of distinct distances from $x$ is at least $cp$. The main purpose of this paper is to provide extensions in a very general structure of pinned trees, which is inspired by the recent work due to Ou and Taylor (2021).
Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an improved bound for the number of dot products determined by $P$, proving that, \[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the proof of this bound is a new superquadratic expander involving products and shifts. We prove that, for any finite set $X \subset \mathbb R$, there exist $z,z' \in X$ such that \[ \left|\frac{(zX+1)^{(2)}(z'X+1)^{(2)}}{(zX+1)^{(2)}(z'X+1)}\right| \gtrsim |X|^{5/2}. \] This is derived from a more general result concerning growth of sets defined via convexity and sum sets, and which can be used to prove several other expanders with better than quadratic growth. The proof develops arguments from recent work by the first two listed authors and Misha Rudnev, and uses predominantly elementary methods.
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 $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime. Specifically, given a large finite set of points $E$, and a sequence of elements of the base field (or ring) $(α_1,\ldots,α_k)$, we give conditions guaranteeing the expected number of $(k+1)$-tuples of distinct points $(x_1,\dots, x_{k+1})\in E^{k+1}$ satisfying $x_j \cdot x_{j+1}=α_j$ for every $1\leq j \leq k$.
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 finite set of $n$ points $E$, and a sequence of angles $(α_1,\ldots,α_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(x_1,\dots, x_{k+2})\in E^{k+2}$ satisfying $\angle (x_j,x_{j+1},x_{j+2})=α_j$ for every $1\le j \le k$ as well as pinned analogues.
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, given a large finite set of $n$ points $E$, we look for bounds on how many subsets of $k$ points satisfy a set of relationships between point pairs based on distances or dot products. We survey some of the recent work in the area and present several new, more general families of bounds.
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 finite set of $n$ points $E$, and a sequence of nonzero dot products $(α_1,\ldots,α_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(A_1,\dots, A_{k+1})\in E^{k+1}$ satisfying $A_j \cdot A_{j+1}=α_j$ for every $1\leq j \leq k$.
We present a family of sharpness examples for Falconer-type single dot product results. In particular, for $d\geq 2,$ for any $s<\frac{d+1}{2},$ we construct a Borel probability measure $μ$ satisfying the energy estimate $I_s(μ)<\infty,$ yet the estimate \begin{equation} (μ\times μ)\{(x,y):1\leq x\cdot y \leq 1+ε\} \leq Cε\end{equation} does not hold with constants independent of $ε$. It is known (\cite{EIT11}) that such an estimate always holds with $C$ independent of $ε$ if $I_{\frac{d+1}{2}}(μ)<\infty$. Thus our estimate proves the sharpness of the dimensional threshold in this result and generalizes similar results (\cite{Mat95}, \cite{IS16}) established in the case when the dot product $x \cdot y$ is replaced by the Euclidean distance function $|x-y|$, or, more generally, ${||x-y||}_K$, the distance that comes from the norm induced by a symmetric convex body $K$ with a smooth boundary and non-vanishing curvature. Our constructions are partially based on ideas that come from discrete incidence theory.
In this paper, we show that the maximum number of points in $d\geq3$ dimensions determining exactly 2 distinct triangles is $2d$. We further show that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We build upon the work of Hirasaka and Shinohara who determined that the $d$-orthoplex is such an optimal configuration, but did not prove its uniqueness. Further, we present a more elementary argument for its optimality.
We study a generalization of Erd\H os's unit distances problem to chains of $k$ distances. Given $\mathcal P,$ a set of $n$ points, and a sequence of distances $(δ_1,\ldots,δ_k)$, we study the maximum possible number of tuples of distinct points $(p_1,\ldots,p_{k+1})\in \mathcal P^{k+1}$ satisfying $|p_j p_{j+1}|=δ_j$ for every $1\leq j \leq k$. We study the problem in $\mathbb R^2$ and in $\mathbb R^3$, and derive upper and lower bounds for this family of problems.
Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2 \leq k \leq d$ and is obtained through counting simplices in a standard lattice using results of the Gauss circle problem. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We also establish a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing an example of Mattila. Finally, we prove a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for triangles in a convex setting in every dimension greater than $3$. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.
We study a variation of the Shepard approximation scheme by introducing a dilation factor into the base function, which synchronizes with the Hausdorff distance between the data set and the domain. The novelty enables us to establish an optimal order Jackson \cite{jackson} type error estimate (with an explicit constant) for bounded continuous functions on any given convex domain. We also improve en route an upper bound estimate due to Narcowich and Ward for the numbers of well-separated points in thin annuli, which is of independent interest.
We seek conditions under which colorings of various vector spaces are guaranteed to have a copy of a unit equilateral triangle, having each vertex in a different color class. In particular, we explore the analogous question in the setting of vector spaces finite fields, with an appropriate notion of distance.
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 power of a prime. More precisely, given $E\subset \mathbb F_q^d$ or $\mathbb Z_q^d$, we provide bounds on the size of the set \[\left\{(u,v,w)\in E \times E \times E : u\cdot v = α, u \cdot w = β\right\}\] for units $α$ and $β$.
Given a large finite point set, $P\subset \mathbb R^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(α, β)$, we bound the size of the set $$\left\{(p,q,r)\in P \times P \times P : p \cdot q = α, p \cdot r = β\right\}.$$
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 elementary, the bounds are, in some sense, best possible. The upper bound is also connected to triangular numbers, and a slight improvement to this bound could be obtained by resolving a problem on them. In the final section, we consider implications of our bounds in various settings, including finite fields, coding theory and additive combinatorics. In particular, we obtain the first non-trivial upper bound for the image set of a planar function over a finite field; this bound is better than the bound implied by the Dembowski-Ostrom conjecture.
We give a lower bound for the number of ordinary lines spanned by a modular hyperbola when the modulus is a prime power. We also give a partial answer to a question of Shparlinski.