arXiv subjects
Publications and source records attributed to Kyle Hambrook.
We prove that, in an arbitrary ordered field, L'H\^{o}pital'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 corollary of L'H\^{o}pital's Rule.
We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions $d$ and the full parameter range $0 < a,b < d$. Our construction is deterministic and also yields Salem sets.
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 $\sum_{q \in \mathbb{Z}^n} Ψ(q)^m < \infty$.
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 Hambrook and {\L}aba \cite{HL2013} and Chen \cite{chen} required randomness and only covered the range $0 < b \leq a \leq d=1$. We also resolve a question of Seeger \cite{seeger-private} about the Fourier restriction inequality on the sets of well-approximable numbers.
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 natural question to ask is whether the same phenomena holds for well approximable vectors in $\mathbb{R}^n.$ We prove that this is in general not the case. In addition, we also show that in $\mathbb{R}^n, n\geq 2,$ the set of badly approximable vectors is not Salem.
We investigate the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sets of the form $RY + Z, $ where $R \subseteq (0,\infty)$ and $Y, Z \subseteq \mathbb{R}^d$. We prove a theorem on the Lebesgue measure and Hausdorff dimension of $RY+Z$; The theorem is a generalized variant of some theorems of Wolff and Oberlin in which $Y$ is the unit sphere, but its proof is much simpler. We also prove a deeper existence theorem: For each $α\in [0,1]$ and for each non-empty compact set $R \subseteq (0,\infty)$, there exists a compact set $Y \subseteq [1,2]$ such that $\dim_F(Y) = \dim_H(Y) = \overline{\dim_M}(Y) = α$ and $\dim_F(RY) \geq \min\{ 1, \dim_F(R) + \dim_F(Y)\}$. This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones.
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) reconstruction quality is deficient; (c) reconstruction quality is contingent on a computationally expensive process of tuning hyperparameters. In this work, we address these issues by utilizing Denoising Auto Encoders (DAEs) as priors and a projected gradient descent algorithm for recovering the original signal. We provide rigorous theoretical guarantees for our method and experimentally demonstrate its superiority over existing state of the art methods in compressive sensing, inpainting, and super-resolution. We find that our algorithm speeds up recovery by two orders of magnitude (over 100x), improves quality of reconstruction by an order of magnitude (over 10x), and does not require tuning hyperparameters.
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 based on a form of Diophantine approximation in number fields.
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 $\ell_{\infty}$-norm attacks. Our results are general as they can be applied to most unitary transforms used in practice and hold for $\ell_0$-norm, $\ell_2$-norm, and $\ell_\infty$-norm bounded noise. In the case of $\ell_0$-norm noise, we prove recovery guarantees for Iterative Hard Thresholding (IHT) and Basis Pursuit (BP). For $\ell_2$-norm bounded noise, we provide recovery guarantees for BP and for the case of $\ell_\infty$-norm bounded noise, we provide recovery guarantees for Dantzig Selector (DS). These guarantees theoretically bolster the defense framework introduced in \cite{bafna2018thwarting} for defending neural networks against adversarial inputs. Finally, we experimentally demonstrate the effectiveness of this defense framework against an array of $\ell_0$, $\ell_2$ and $\ell_\infty$ norm attacks.
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 satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the $p$-adic setting.
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 $E \subseteq \mathbb{R}^d$, we define $$ Δ_{\textbf{d}}(E) = \left\{ \left(|x^{(1)}-y^{(1)}|,\ldots,|x^{(\ell)}-y^{(\ell)}|\right) : x,y \in E \right\} \subseteq \mathbb{R}^{\ell}, $$ where for $x\in \mathbb{R}^d$ we write $x=\left( x^{(1)},\dots, x^{(\ell)} \right)$ with $x^{(i)} \in \mathbb{R}^{d_i}$. We ask how large does the Hausdorff dimension of $E$ need to be to ensure that the $\ell$-dimensional Lebesgue measure of $Δ_{\textbf{d}}(E)$ is positive? We prove that if $2 \leq d_i$ for $1 \leq i \leq \ell$, then the conclusion holds provided $$ \dim(E)>d-\frac{\min d_i}{2}+\frac{1}{3}.$$ We also note that, by previous constructions, the conclusion does not in general hold if $$\dim(E)<d-\frac{\min d_i}{2}.$$ A group action derivation of a suitable Mattila integral plays an important role in the argument.
We construct explicit (i.e., non-random) examples of Salem sets in $\mathbb{R}^2$ of dimension $s$ for every $0 \leq s \leq 2$. In particular, we give the first explicit examples of Salem sets in $\mathbb{R}^2$ of dimension $0 < s < 1$. This extends a theorem of Kaufman.
We prove the range of exponents in the general $L^2$ Fourier restriction theorem due to Mockenhaupt, Mitsis, Bak and Seeger is sharp for a large class of measures on $\mathbb{R}^d$. This extends to higher dimensions the sharpness result of Hambrook and Łaba.
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 $$ E(Q,Ψ,θ) = \{ x \in \mathbb{R} : \| q x - θ\| \leq Ψ(q) \text{ for infinitely many $q \in Q$} \}. $$ We prove a lower bound on the Fourier dimension of $E(Q,Ψ,θ)$. This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimension of $E(Q,Ψ,θ)$ in new cases. We also prove a higher-dimensional analog of our result.
We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λand a Dirichlet character χ. As a consequence of this result we prove an effective variant of the Bombieri-Vinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes.
We prove that the range of exponents in Mockenhaupt's restriction theorem for Salem sets, with the endpoint estimate due to Bak and Seeger, is optimal.