As a consequence of the radial projection theorems of Orponen-Shmerkin-Wang and Ren, both groups of authors obtain a continuum Beck-type theorem for lines.
In particular, they show that if a Borel set \(X\subset \mathbb{R}^n\) is not too concentrated on \(m\)-planes (i.e. \(\dim (X\setminus P^m) = \dim X\) for all \(m\)-planes \(P^m\)), then the set of lines containing at least 2 points of \(X\), \(\mathcal L(X)\), satisfies \(\dim \mathcal L(X) \geq \min\{2\dim X, 2m\}\).
Note that my work with Marshall extends this lowerbound in a reasonable way to sets \(X\) such that \(\dim (X\setminus P) < \dim X\) for some \(m\)-plane \(P\), but such that \(\dim (X\setminus P^m) \geq t\) for all \(m\)-planes \(P^m\) and some \(0 < t <\dim X\).
In this joint work with Alex Ortiz and Dmitrii Zakharov, we obtain a sharp Beck-type theorem for hyperplanes. Say that a set \(X\subset \mathbb{R}^n\) is nonconcentrated, or simply NC, if
\[
\dim \left(X\setminus \bigcup V_i \right) = \dim X \quad \text{ for all flats \{V_i\} such that } \quad \sum\dim V_i \leq n-1.
\]
Then, we show that the set of affine hyperplanes that contain \(n\)-affinely independent points of \(X\), \(\mathcal P^{n-1}(X)\), satisfies
\[
\dim \mathcal P^{n-1}(X) \geq n \min\{\dim X, 1\}.
\]
This bound is sharp in two natural senses. Firstly, if there exists a collection of flats \(\{V_i\}\) such that \(\sum \dim V_i \leq n-1\) and \(X\subset \bigcup V_i\), it's a short argument using Lipschitz properties to show that
\[
\dim \mathcal P^{n-1}(X) \leq (n-2) \min\{\dim X, 1\}.
\]
In this sense, it is sharp in the geometric assumptions. Secondly, another Lipschitz argument quickly shows that
\[
\dim \mathcal P^{n-1}(X) \leq n \min\{\dim X, 1\}
\]
and thus the bound is sharp numerically. In the discrete setting, this is saying that a finite set of \(N\) points \(X\subset \mathbb{R}^n\) satisfies \(|\mathcal P^{n-1}(X)| \leq |X|^{n}\). Both of these arguments are included in the introduction of the paper.
The key ingredient in our approach is to generalize the thin-tubes methodology of Orponen-Shmerkin-Wang and Ren to thin \(k\)-planes.
The definition is slightly different than that of thin tubes (in particular, thin 1-planes is not by definition the same as thin tubes), but one can derive the existence of one from the other up to irrelevant constants (see the Appendix of the paper).
Furthermore, the NC condition we consider is motivated by combinatorial results due to Do and Lund, with our measure theoretic approach being largely motivated by Do's argument.
My UBC Master's thesis, completed under the cosupervision of Izabella Łaba, Pablo Shmerkin, and Josh Zahl. This thesis gives an exposition into a number of research topics in projection theory, including orthogonal projections, Furstenberg sets, and radial projections, using discrete geometry as motivation. The thesis concludes with a discussion of two nice applications of projection theory: Beck-type theorems (joint with Marshall), and pinned dot product sets (joint with Marshall and Senger).
We study a variant of the Furstenberg set problem for high dimensional flats in Euclidean space. The "standard" Furstenberg set problem asks the following: Given \(X\subset \mathbb{R}^n\) Borel, suppose there exists a nonempty \(t\)-dimensional set of affine \(k\)-planes \(\mathcal{P}\) such that for each \(P \in \mathcal P\), \(\dim (X\cap P) \geq s\). How large must \(\dim X\) be in terms of \(s,t,n,\) and \(k\)? Such sets are referred to as \((s,t;k)\)-Furstenberg sets. The variant we study involves \((s,t;k)\)-spread Furstenberg sets, defined as follows: Given \(X\subset \mathbb{R}^n\) Borel, suppose there exists a nonempty \(t\)-dimensional set of \(k\)-dimensional subspaces \(\mathcal{P}\) such that for each \(P \in \mathcal P\) there exists a translation vector \(a_P\in \mathbb{R}^n\) such that \(\dim (X\cap (P + a_P)) \geq s\). How large must \(\dim X\) be in terms of \(s,t,n,\) and \(k\)? The key difference here is that the set of \(k\)-dimensional flats are "spread out" in different directions (by assuming they come from translatins of subspaces). We find lower bounds for \((s,t;k)\)-spread Furstenberg sets over Euclidean space, following the methodology of Dhar-Dvir-Lund over finite fields. This also raises interesting questions regarding the relationship between the original problem and this given variant discussed within the paper.
We study a variant of the Falconer distance problem for dot products. In particular, given \(A\subset \mathbb{R}^n\) Borel, and \(a,x\in \mathbb{R}^n\), we study sets of the form \(\Pi_x^a(A) := \{(a-x)\cdot y : y\in A\}\) where \(\cdot\) here is the standard dot product over \(\mathbb{R}^n\). We are able to show that for \(\dim A\) sufficiently large, \(\Pi_x^a(A)\) must be large for many \(a\in A\), where here "largeness" is in terms of positive measure, large Hausdorff dimension, or non-empty interior. Interestingly, our approach for all three senses of the word "large" is identical, making use of exceptional set estimates (for orthogonal projections) and radial projection estimates (both classical and recent).
Let \(X \subset \mathbb{R}^n\) be a Borel set, and consider the set of lines, \(\mathcal{L}(X)\subset \mathcal {A}(n,1)\), containing at least 2 points in \(X\). In recent work of Orponen-Shmerkin-Wang and Ren, radial projection estimates and (dual) Furstenberg set bounds have been used to lowerbound \(\dim \mathcal{L}(X)\) in terms of \(\dim X\). Such work is seen as a continuum version of Beck's theorem from point-line incidence geometry. Utilizing results of B.-Fu-Ren, together with Caleb Marshall, we find a continuum Erdős-Beck theorem. We motivate such work via their discrete counterparts, and discuss a new conjecture regarding the dimension of line sets.
We generalize recent work of Orponen-Shmerkin-Wang using two different methods. In particular, we show the following bilinear radial projection estimates: Given \(X,Y\subset \mathbb{R}^n\) Borel sets and \(X\neq 0\). If \(\dim Y\in (k,k+1]\) for some \(k \in \{1,\dots,n-1\}\), then \[\sup_{x\in X} \dim \pi_x(Y\setminus \{x\}) \geq \min\{\dim X + \dim Y - k,k\}.\] The first method is shorter, and proves the equivalence between this result and a result of B.-Gan/OSW. 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.
This study guide on recent work of Orponen-Shmerkin-Wang was written during the UPenn Study Guide Writing Workshop 2023 under the mentorship of Josh Zahl. We expound upon the major themes and clarify technical details of the paper "Kaufman and Falconer estimates for radial projections and a continuum version of Beck's theorem" of Orponen-Shmerkin-Wang.
We study exceptional set estimates for orthogonal projections over finite fields, \(\mathbb{F}_q^n\). In particular, given \(A\subset \mathbb{F}_q^n\) with \(|A| = q^a\), and \(0< s < \min(k,a)\), we study sets of the form \[ E(A) := \{V\in G(k,\mathbb{F}_q^n) : \# \pi_V(A) < q^s\}. \] Such sets are known as exceptional set estimates. We also show that the bound \(|E(A)| \lesssim p^{2s-a}\), conjectured to be true for all \(A\subset \mathbb{F}_p^n\) with \(|A| = q^a\), is sharp (if \(p=q\) is prime).
This research was conducted during SPUR (the Summer Program for Undergraduate Research in mathematics) 2022 at MIT, and was published in Annales Fennici Mathematici in 2024.
We study exceptional set estimates for radial and orthogonal projections over Euclidean space, \(\mathbb{R}^n\). Given a Borel set \(A\subset \mathbb{R}^n\), with \(\dim A = a\), how often is the shadow of \(A\) onto a \(m\)-dimensional subspace, \(V\in G(n,m)\) large?
Marstrand's projection theorem, which states that for almost every \(V\in G(n,m)\), \(\dim \pi_V(A) = \min\{a,m\}\).
Exceptional set estimates try to quantify this statement further. In particular, given \(0 < s \leq \min\{a,m\}\),
how can we bound
\[
\dim \{V\in G(n,m) : \dim \pi_V(A)< s \}?
\]
Two such bounds were proven by Kaufman and Falconer. The first half of this SPUR project involved reproving the results of
Kaufman and Falconer via delta-discretization, a high-low argument from Fourier analysis, and a counting argument.
One can study similar types of exceptional set estimates for radial projections. Utilizing a similar high-low argument from
the first half of the project, we proved two open conjectures by Lund-Pham-Thu and Liu. The statements of said radial projection conjectures were first proven in \(\mathbb{R}^2\)
by Orponen-Shmerkin, and we generalized these statements to \(\mathbb{R}^n\) following a similar framework.
Notably, using different methods, Orponen-Shmerkin-Wang proved the radial projection statements proven in this paper shortly after this paper was released.
See slides linked above to see more sources/statements regarding exceptional set estimates for orthogonal and radial projections.
The work of B.-Gan (see above) was motivated by radial projection conjectures due to Lund-Pham-Thu and Liu. My initial work with Shengwen Gan proved the conjectures over Euclidean space, while Lund-Pham-Thu's conjecture was initially stated over finite fields. Hence, in this work with Ben Lund and Thang Pham, we prove the finite field radial projection conjectures.
This research was conducted during Williams College's SMALL REU 2023 (Research Experience for Undergraduates) under the mentorship of Alex Iosevich and Eyvi Palsson.
One can study point configuration problems over finite fields. See the explanation in the next bullet point for further detail.
In the previous paper on embedding certain configurations (see below), we studied embeddings with a "dot product" distance.
In this work, we generalize the statements to distances defined via a general non-degenerate bilinear form or quadratic form.
This research was conducted during Williams College's SMALL REU 2023 (Research Experience for Undergraduates) under the mentorship of Alex Iosevich and Eyvi Palsson.
One can study point configuration problems over finite fields in the following sense.
Let \(G\) be a connected graph, and assign weights to each edge in \(\mathbb{F}_q\).
We call such graphs distance graphs. We study the following problem:
Given \(E\subset \mathbb{F}_q^d\), how large does \(|E|\) need to be until it contains at least
one isometric copy of the distance graph \(G\)?
We study this problem for specific types of distance graphs, with a "distance"
defined over \(\mathbb{F}_q^n\) being (roughly) given by the standard dot product mod \(q\).
This builds upon work of Iosevich-Parshall.
I typed lecture notes for the Matrix Calculus class (2023) taught by Alan Edelman and Steven G. Johnson. This course discusses how the ideas from multivariable calculus generalizes to matrices.
I created and taught a bridge class between two different real analysis courses at MIT.
The first, 18.100A/18.100P, focusses on real analysis over Euclidean sapces \(\mathbb{R}^n\), and
the latter, 18.100B/18.100Q, focusses on real analysis over metric spaces.
I obtained the MIT Mathematics Department Teaching and Learning Award for this work, and the course's
material is hosted on OpenCourseWare.
I guest hosted for OCW's podcast Chalk Radio. In this podcast, I interview Haynes Miller about communication and mathematics. I created and taught a bridge class between two different real analysis courses at MIT.
Typed lecture notes for 18.100A on OCW, taught by Casey Rodriguez. The course covered real analysis over Euclidean spaces.
A discussion on student contributions to open educational resources (OERs), in particular with respect to OpenCourseWare (OCW).