The Heil-Ramanathan-Topiwala (HRT) conjecture is an open problem in time-frequency analysis. It asserts that any finite combination of time-frequency shifts of a non-zero function in $L^2(\mathbb{R})$ is linearly independent. Despite its simplicity, the conjecture remains unproven in full generality, with only specific cases resolved.
The Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem extends the classical restriction theorem for measures on smooth manifolds to fractal measures. We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions. The proof uses number fields to construct fractal measures in R^d. This work is joint with Robert Fraser and Kyle Hambrook.
Kakeya sets are compact subsets of $\mathbb{R}^n$ that contain a unit line segment pointing in every direction and the Kakeya conjecture states that such sets must have Hausdorff dimension $n$. The property of stickiness was first discovered by by Katz-Laba-Tao in their 1999 breakthrough paper on the Kakeya problem. Then Wang-Zahl formalized the definition of a sticky Kakeya set as a subclass of general Kakeya sets in 2022.
The Heisenberg projection problem asks whether there is an analogue of the Marstrand projection theorem in the first Heisenberg group, namely whether Hausdorff dimension of sets generically decreases under projection, for a natural family of projections arising from the group structure. This problem is still open, but I will discuss a recent improvement to the known bound obtained through a variable coefficient local smoothing inequality.
We will present advances on the boundedness of geometric maximal operators, focusing on a recent result from joint work with Paul Hagelstein and Alex Stokolos, which employs probabilistic techniques in the construction of Kakeya-type sets. The material presented extends ideas of M. Bateman and N. Katz.
The `middle third integer Cantor set' consists of those integers which do not have a 2 in their base 3 representation. We will review and extend some results about such sets. For a general integer Cantor set K, with 0 as an allowed digit, it is known that K is intersective, a result of Furstenberg-Katznelson.
We first investigate reproducing pairs in Hilbert spaces, with a focus on the discrete case. Reproducing pairs generalize frames and consist of two sequences $\Psi$ and $\Phi$, along with a bounded invertible operator $S_{\Psi,\Phi}$. The work examines sequences that are overcomplete by one element—that is, they become exact upon removal of a single element. A central result shows that if such a sequence admits a reproducing partner, the resulting exact subsequence must form a Schauder basis.
In this thesis, we investigate three related problems at the intersection of analytic number theory and discrete harmonic analysis. Our primary goal is to understand discrete averaging operators over arithmetic sets—discrete analogues of classical continuous operators—and analyze their behavior using tools from harmonic analysis and additive combinatorics. The results deepen our understanding of how analytic and combinatorial techniques interact in the study of primes and other arithmetic structures.
An important class of problems at the intersection of harmonic analysis and geometric measure theory asks how large the Hausdorff dimension of a set must be to ensure that it contains certain types of geometric point configurations. We apply these tools to study configurations associated to the problem of bounding the VC-dimension of a naturally arising class of indicator functions on fractal sets.
For \( c\in(1,2) \) we consider the following operators
\[
\mathcal{C}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor}}{n} \bigg|\text{,}
\]
\[
\mathcal{C}^{\mathsf{sgn}}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \mathsf{sign}(n) \lfloor |n|^{c} \rfloor}}{n} \bigg| \text{,}
\]
and prove that both extend boundedly on \( \ell^p(\mathbb{Z}) \), \( p\in(1,\infty) \).
