Kyle Hambrook
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View article: L'Hôpital's Rule is Equivalent to the Least Upper Bound Property
L'Hôpital's Rule is Equivalent to the Least Upper Bound Property Open
We prove that, in an arbitrary ordered field, L'Hôpital's Rule is true if and only if the Least Upper Bound Property is true. We do the same for Taylor's Theorem with Peano Remainder, and for one other property sometimes given as a corolla…
View article: Fourier restriction and well-approximable numbers
Fourier restriction and well-approximable numbers Open
We use a deterministic construction to prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for dimension $$d=1$$ and parameter range $$0 < a,b \le d$$ and $$b\le 2a$$ . …
View article: On the Exact Fourier Dimension of Sets of Well-Approximable Matrices
On the Exact Fourier Dimension of Sets of Well-Approximable Matrices Open
We compute the exact Fourier dimension of the set of $Ψ$-well-approximable $m \times n$ matrices (and the set of $Ψ$-well-approximable numbers) in the homogeneous and inhomogeneous cases for any approximation function $Ψ$ satisfying $\sum_…
View article: Fourier restriction and well-approximable numbers
Fourier restriction and well-approximable numbers Open
We use a deterministic construction to prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for dimension $d=1$ and parameter range $0 < a,b \leq d$ and $b\leq 2a$. Previous constructions by…
View article: The Perceived Impact of Faculty-in-Residence Programs on Faculty Development
The Perceived Impact of Faculty-in-Residence Programs on Faculty Development Open
Faculty-in-Residence (FIR) programs, where students interact with faculty outside of the classroom, have shown positive effects on student success. However, most research does not look at FIR programs from a holistic perspective that exami…
View article: Non-Salem Sets in Metric Diophantine Approximation
Non-Salem Sets in Metric Diophantine Approximation Open
A classical result of Kaufman states that, for each $\tau>1$, the set of $\tau $-well approximable numbers $$ \begin{align*} & E(\tau)=\{x \in \mathbb{R}: |xq-r| < |q|^{-\tau} \text{ for infinitely many integer pairs } (q,r)\} \end{a…
View article: DAEs for Linear Inverse Problems: Improved Recovery with Provable Guarantees
DAEs for Linear Inverse Problems: Improved Recovery with Provable Guarantees Open
Generative priors have been shown to provide improved results over sparsity priors in linear inverse problems. However, current state of the art methods suffer from one or more of the following drawbacks: (a) speed of recovery is slow; (b)…
View article: Non-Salem sets in metric Diophantine approximation
Non-Salem sets in metric Diophantine approximation Open
A classical result of Kaufman states that, for each $τ>1,$ the set of well approximable numbers \[ E(τ)=\{x\in\mathbb{R}: \|qx\| < |q|^{-τ} \text{ for infinitely many integers q}\} \] is a Salem set with Hausdorff dimension $2/(1+τ)$. A na…
View article: DAEs for Linear Inverse Problems: Improved Recovery with Provable Guarantees
DAEs for Linear Inverse Problems: Improved Recovery with Provable Guarantees Open
Generative priors have been shown to provide improved results over sparsity priors in linear inverse problems. However, current state of the art methods suffer from one or more of the following drawbacks: (a) speed of recovery is slow; (b)…
View article: Explicit Salem sets in $\mathbb{R}^n$
Explicit Salem sets in $\mathbb{R}^n$ Open
We construct the first explicit (i.e., non-random) examples of Salem sets in $\mathbb{R}^n$ of arbitrary prescribed Hausdorff dimension. This completely resolves a problem proposed by Kahane more than 60 years ago. The construction is base…
View article: Recovery Guarantees for Compressible Signals with Adversarial Noise
Recovery Guarantees for Compressible Signals with Adversarial Noise Open
We provide recovery guarantees for compressible signals that have been corrupted with noise and extend the framework introduced in \cite{bafna2018thwarting} to defend neural networks against $\ell_0$-norm, $\ell_2$-norm, and $\ell_{\infty}…
View article: Explicit Salem sets and applications to metrical Diophantine approximation
Explicit Salem sets and applications to metrical Diophantine approximation Open
Let $Q$ be an infinite subset of $\mathbb{Z}$, let $Ψ: \mathbb{Z} \rightarrow [0,\infty)$ be positive on $Q$, and let $θ\in \mathbb{R}$. Define $$ E(Q,Ψ,θ) = \{ x \in \mathbb{R} : \| q x - θ\| \leq Ψ(q) \text{ for infinitely many $q \in Q$…
View article: Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the $p$-adic numbers
Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the $p$-adic numbers Open
We exhibit the first explicit examples of Salem sets in $\mathbb{Q}_p$ of every dimension $0 < α< 1$ by showing that certain sets of well-approximable $p$-adic numbers are Salem sets. We construct measures supported on these sets that sati…
View article: Explicit Salem sets, Fourier restriction, and metric Diophantine\n approximation in the $p$-adic numbers
Explicit Salem sets, Fourier restriction, and metric Diophantine\n approximation in the $p$-adic numbers Open
We exhibit the first explicit examples of Salem sets in $\\mathbb{Q}_p$ of\nevery dimension $0 < \\alpha < 1$ by showing that certain sets of\nwell-approximable $p$-adic numbers are Salem sets. We construct measures\nsupported on these set…
View article: Group actions and a multi-parameter Falconer distance problem
Group actions and a multi-parameter Falconer distance problem Open
In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given ${\textbf{d}}=(d_1,d_2, \dots, d_{\ell})\in \mathbb{N}^{\ell}$ with $d_1+d_2+\dots+d_{\ell}=d$ and $E \subseteq \mathbb{R}^d$, …
View article: Sharpness of the Mockenhaupt–Mitsis–Bak–Seeger restriction theorem in higher dimensions
Sharpness of the Mockenhaupt–Mitsis–Bak–Seeger restriction theorem in higher dimensions Open
We prove the range of exponents in the general $L^2$ Fourier restriction\ntheorem due to Mockenhaupt, Mitsis, Bak and Seeger is sharp for a large class\nof measures on $\\mathbb{R}^d$. This extends to higher dimensions the sharpness\nresul…