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Paige Bright

Publications and source records attributed to Paige Bright.

14 recordsLinked to original sources
arXiv2026

Applications of Nonlinear Projections to Rectifiable 1-sets

Projection theorems in Euclidean space provide a fundamental link between the geometric structure of a set and the size of its lower-dimensional images. For 1-rectifiable sets in $\mathbb{R}^d$, a classical theorem of Federer shows that the 1-dimensional Hausdorff measure of such sets is controlled by the multiplicity-weighted lengths of finitely many linearly independent projections. We develop a framework for extending Federer's result into a diverse set of nonlinear problems. This technique yields a unified approach for studying sets through their lower-dimensional nonlinear images, as well as studying the exceptional sets which exhibit poor projective behavior. As illustrations of our technique, we show that (i) every 1-rectifiable set contains a pin whose pinned distance set has positive Lebesgue measure, and that the exceptional set of pins for which this fails is contained in a $(d-2)$-dimensional affine subspace; (ii) planar radial projections of a 1-rectifiable set can fail to have positive length from at most one vantage point unless the set is essentially linear; and finally (iii) unions of circles centered on a 1-rectifiable set have positive area under mild assumptions on the radius function.

math.CA↗
arXiv2025

A Continuum Beck-type Theorem for Hyperplanes

We prove a sharp continuum Beck-type theorem for hyperplanes. Our work is inspired by foundational work of Beck on the discrete problem, as well as refinements due to Do and Lund. The inductive proof uses recent breakthrough results in projection theory by Orponen--Shmerkin--Wang and Ren, who proved continuum Beck-type theorems for lines in $\mathbb{R}^2$ and $\mathbb{R}^n$.

math.CA↗
arXiv2025

Progress in Projection Theory and other dimensional developments

We provide exposition into the field of projection theory, which lies at the intersection of incidence geometry and geometric measure theory. We first give the necessary preliminaries in Chapter 2, focusing on incidences between points and lines and the definition of Hausdorff dimension. With this background in tow, in Chapter 3 we dive into thorough surveys on three topics in projection theory: orthogonal projections, Furstenberg sets, and radial projections. We particularly highlight the interconnectedness of these topics in both the discrete and continuum settings. Through these surveys, we also give the necessary background for discussing applications of projection theory in Chapter 4. The first application is on Beck-type problems, first studied in the discrete setting by J\'ozsef Beck in 1983 and in the continuum setting by Orponen, Shmerkin, and Wang in 2022. Given $X\subset \mathbb{R}^n$, these problems seek to understand how large the set of lines that contain at least 2 points of $X$, $\mathcal L(X)$, can be. To this end, we present a continuum Erd\H{o}s--Beck theorem due to myself and Marshall in 2024, which motivates and makes use of a dual Furstenberg set estimate due to myself, Fu, and Ren from the same year. The second application is on Falconer-type (distance) problems which have been a prominent topic in both the discrete and continuum settings. Given $X\subset \mathbb{R}^n$, these problems seek to understand how large $X$ must be until the set of distinct "distances" between points of $X$ is large (for a reasonable notion of "distance"). To this end, we present a Falconer-type distance problem for dot products due to myself, Marshall, and Senger in 2024, making use of both standard and modern results for orthogonal and radial projections.

math.CA↗
arXiv2025

Matrix Calculus (for Machine Learning and Beyond)

This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and return a matrix inverse or factorization, derivatives of ODE solutions, and even stochastic derivatives of random functions. It emphasizes practical computational applications, such as large-scale optimization and machine learning, where derivatives must be re-imagined in order to be propagated through complicated calculations. The class also discusses efficiency concerns leading to "adjoint" or "reverse-mode" differentiation (a.k.a. "backpropagation"), and gives a gentle introduction to modern automatic differentiation (AD) techniques.

math.HO↗
arXiv2024

Spread Furstenberg Sets

We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\mathcal P \subset \mathcal G(n,k)$ such that for all $P\in \mathcal P$, there exists a translation vector $a_P \in \mathbb{R}^n$ such that $\dim(F\cap (P + a_P)) \geq s$. We show that given $k \geq k_0 +1$ (where $k_0:= k_0(n)$ is sufficiently large) and $s>k_0$, every $(s,t;k)$-spread Furstenberg set $F$ in $\mathbb{R}^n$ satisfies \[ \dim F \geq n-k + s - \frac{k(n-k) - t}{\lceil s\rceil - k_0 +1 }. \] Our methodology is motivated by the work of the second author, Dvir, and Lund over finite fields.

math.CA↗
arXiv2024

Pinned Dot Product Set Estimates

We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\subset \mathbb{R}^n$ and $a,x\in \mathbb{R}^n$, we study sets of the form \[ \Pi_x^a(A) := \{\alpha \in \mathbb{R} : (a-x)\cdot y= \alpha, \text{ for some $y\in A$}\}. \] We discuss some of what is already known to give a picture of the current state of the art, as well as prove some new results and special cases. We obtain lower bounds on the Hausdorff dimension of $A$ to guarantee that $\Pi^a_x(A)$ is large in some quantitative sense for some $a\in A$ (i.e. $\Pi_x^a(A)$ has large Hausdorff dimension, positive measure, or nonempty interior). Our approach to all three senses of "size" is the same, and we make use of both classical and recent results on projection theory.

math.CA↗
arXiv2024

Radial Projections in $\mathbb{R}^n$ Revisited

We generalize the recent results on radial projections by Orponen, Shmerkin, Wang using two different methods. In particular, we show that given $X,Y\subset \mathbb{R}^n$ Borel sets and $X\neq \emptyset$. If $\dim Y \in (k,k+1]$ for some $k\in \{1,\dots, n-1\}$, then \[ \sup_{x\in X} \dim \pi_x(Y\setminus \{x\}) \geq \min \{\dim X + \dim Y - k, k\}. \] Our results give a new approach to solving a conjecture of Lund-Pham-Thu in all dimensions and for all ranges of $\dim Y$. The first of our two methods for proving the above theorem is shorter, utilizing a result of the first author and Gan. Our second method, though longer, follows the original methodology of Orponen--Shmerkin--Wang, and requires a higher dimensional incidence estimate and a dual Furstenberg-set estimate for lines. These new estimates may be of independent interest.

math.CA↗
arXiv2024

A Continuum Erd\H{o}s-Beck Theorem

We prove a version of the Erd\H{o}s--Beck Theorem from discrete geometry for fractal sets in all dimensions. More precisely, let $X\subset \mathbb{R}^n$ Borel and $k \in [0, n-1]$ be an integer. Let $\dim (X \setminus H) = \dim X$ for every $k$-dimensional hyperplane $H \in \mathcal{A}(n,k)$, and let $\mathcal L(X)$ be the set of lines that contain at least two distinct points of $X$. Then, a recent result of Ren shows $$ \dim \mathcal{L}(X) \geq \min \{2 \dim X, 2k\}. $$ If we instead have that $X$ is not a subset of any $k$-plane, and $$ 0<\inf_{H \in \mathcal{A}(n,k)} \dim (X \setminus H) = t < \dim X, $$ we instead obtain the bound $$ \dim \mathcal{L}(X) \geq \dim X + t. $$ We then strengthen this lower bound by introducing the notion of the "trapping number" of a set, $T(X)$, and obtain \[ \dim \mathcal L(X) \geq \max\{\dim X + t, \min\{2\dim X, 2(T(X)-1)\}\}, \] as consequence of our main result and of Ren's result in $\mathbb{R}^n$. Finally, we introduce a conjectured equality for the dimension of the line set $\mathcal{L}(X)$, which would in particular imply our results if proven to be true.

math.CA↗
arXiv2023

On a radial projection conjecture and pinned directions in finite spaces

We give upper bounds on the number of exceptional radial projections of arbitrary subsets of vector spaces over finite fields. Our bounds do not depend on the dimension of the ambient space. Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over $\mathbb{F}_q$, let $k \in \{1,2,\ldots,d-1\}$, and let $E \subseteq \mathbb{F}_q^d$ be an arbitrary set of points. We prove two results. First, if $q^{k-1} < |E| \leq 100^{-1}q^{k}$, then the number of points $y$ such that the projection of $E$ from $y$ contains fewer than $50^{-1}|E|$ points is bounded above by $40q^k$. This establishes a conjecture of Lund, Pham, and Thu. Second, if $30q^{k} \leq |E| \leq q^{k+1}$, then the number of points $y$ such that the projection of $E$ from $y$ contains fewer than $M \leq 4^{-1}q^k$ points is bounded above by $300q^kM|E|^{-1}$. We also have an application to a pinned directions problem. Specifically, if $E\subset \mathbb{F}_q^d$ with $|E| > 30q^k$, then there is a point $y \in E$ such that the set of lines incident to $y$ and at least one other point of $E$ determines $q^k/4$ distinct slopes.

math.CO↗
arXiv2023

Generalized point configurations in ${\mathbb F}_q^d$

In this paper, we generalize \cite{IosevichParshall}, \cite{LongPaths} and \cite{cycles} by allowing the \emph{distance} between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of $\F_q^d$ for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting.

math.CO↗
arXiv2023

Improved bounds for embedding certain configurations in subsets of vector spaces over finite fields

The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C q^{m+\frac{d-1}{2}}$, for any $t>0$, there exist $k+1$ points $x^1, \dots, x^{k+1}$ in $E$ such that $||x^i-x^j||=t$ if the $i$'th vertex is connected to the $j$'th vertex by an edge in $G$. In this paper, we give several indications that the maximum degree is not always the right notion of complexity and prove several concrete results to obtain better exponents than the Iosevich-Parshall result affords. This can be viewed as a step towards understanding the right notion of complexity for graph embeddings in subsets of vector spaces over finite fields.

math.CO↗
arXiv2023

Exceptional set estimates in finite fields

We study the exceptional set estimate for projections in $\mathbb{F}_q^n$. For each $V\in G(k,\mathbb{F}^n_q)$, let $$ \pi_V: \mathbb{F}_q^n\rightarrow V $$ be the projection map. We prove the following result: If $A\subset \mathbb{F}_q^n$ with $\#A=q^a$ ($n-1\le a\le n$) and $0< s<\frac{a+n-2}{2}$, then $$ \# \{V\in G(n-1,\mathbb{F}^n_q): \#\pi_V(A)< q^s \}\lessapprox q^{n-2}.$$ This improves the previous range $0 \frac{a+n-2}{2}$, then the right hand side above should be at least $q^t$ for some $t>n-2$.

math.CA↗
arXiv2022

Exceptional set estimates for radial projections in $\mathbb{R}^n$

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set $A\subset \mathbb{R}^n$ such that $\dim A\in (k,k+1]$ for some $k\in\{1,\dots,n-1\}$. For $0<s<k$, we have \[ \text{dim}(\{y\in \mathbb{R}^n \setminus A\mid \text{dim} (\pi_y(A)) < s\})\leq \max\{k+s -\dim A,0\}. \] The second conjecture is by Liu: Given a Borel set $A\subset \mathbb{R}^n$, then \[ \text{dim} (\{x\in \mathbb{R}^n \setminus A \mid \text{dim}(\pi_x(A))<\text{dim} A\}) \leq \lceil \text{dim} A\rceil. \]

math.CA↗