Alex Iosevich
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View article: An approach to endpoint problems in oscillatory singular integrals
An approach to endpoint problems in oscillatory singular integrals Open
In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from $$L^1(\mathbb {R})$$ to $$L^{1,\infty }(\mathbb {R})$$ . The methods we use are entirely elementary, and…
View article: Orlicz spaces and the uncertainty principle
Orlicz spaces and the uncertainty principle Open
Let $f$ be a finite signal. The classical uncertainty principle tells us that the product of the support of $f$ and the support of $\hat{f}$, the Fourier transform of $f$, must satisfy $|supp(f)|\cdot|supp(\hat{f})|\geq |G|$. Recently, Ios…
View article: Some Results in Spectral Synthesis Over ${\mathbb Z}_N^d$
Some Results in Spectral Synthesis Over ${\mathbb Z}_N^d$ Open
A classical result due to Agranovsky and Narayanan (\cite{AN04}) says that if the support of the Fourier transform of $f: {\mathbb R}^n \to {\mathbb C}$ is carried by a smooth measure on a $d$-dimensional manifold $M$, and $f \in L^p({\mat…
View article: Integer distances in vector spaces over finite fields
Integer distances in vector spaces over finite fields Open
The Erdős-Anning Theorem states that an integer distance set in the Euclidean plane must have all of its points on a single line or is finite. However, this is not true if we consider area sets. That is, if \((x_1,y_1)\) and <…
View article: Fourier minimization and imputation of time series
Fourier minimization and imputation of time series Open
One of the most common procedures in modern data analytics is filling in missing values in times series. For a variety of reasons, the data provided by clients to obtain a forecast, or other forms of data analysis, may have missing values,…
View article: Realizing trees of configurations in thin sets
Realizing trees of configurations in thin sets Open
View article: VC-Dimension of Hyperplanes Over Finite Fields
VC-Dimension of Hyperplanes Over Finite Fields Open
Let $$\mathbb {F}_q^d$$ be the d -dimensional vector space over the finite field with q elements. For a subset $$E\subseteq \mathbb {F}_q^d$$ and a fixed nonzero $$t\in \mathbb {F}_q$$ , let $$\mathcal {H}_t(E)=\{h_y: y…
View article: An Approach To Endpoint Problems in Oscillatory Singular Integrals
An Approach To Endpoint Problems in Oscillatory Singular Integrals Open
In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from $L^1(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$. The methods we use are entirely elementary, and rely only on pigeonholing an…
View article: Uncertainty Principle, annihilating pairs and Fourier restriction
Uncertainty Principle, annihilating pairs and Fourier restriction Open
Let $G$ be a locally compact abelian group, and let $\widehat{G}$ denote its dual group, equipped with a Haar measure. A variant of the uncertainty principle states that for any $S \subset G$ and $\Sigma \subset \widehat{G}$, there exists …
View article: Effective support, Dirac combs, and signal recovery
Effective support, Dirac combs, and signal recovery Open
Let $f: {\mathbb Z}_N^d \to {\mathbb C}$ be a signal with the Fourier transform $\widehat{f}: \Bbb Z_N^d\to \Bbb C$. A classical result due to Matolcsi and Szucs (\cite{MS73}), and, independently, to Donoho and Stark (\cite{DS89}) states i…
View article: Buffon Needle Problem Over Convex Sets
Buffon Needle Problem Over Convex Sets Open
We solve a variant of the classical Buffon Needle problem. More specifically, we inspect the probability that a randomly oriented needle of length $l$ originating in a bounded convex set $X\subset\mathbb{R}^2$ lies entirely within $X$. Usi…
View article: Fourier Uncertainty Principles on Riemannian Manifolds
Fourier Uncertainty Principles on Riemannian Manifolds Open
The purpose of this paper is to develop a Fourier uncertainty principle on compact Riemannian manifolds and contrast the underlying ideas with those arising in the setting of locally compact abelian groups. The key obstacle is the growth o…
View article: Packing sets under finite groups via algebraic incidence structures
Packing sets under finite groups via algebraic incidence structures Open
Let $E$ be a set in $\mathbb{F}_p^n$ and $S$ be a set of maps from $\mathbb{F}_p^n$ to $\mathbb{F}_p^n$. We define \[ S (E) := \bigcup_{f\in S} f(E) = \left\lbrace f(x) \colon x\in E, f\in S \right\rbrace.\] In this paper, we establish sha…
View article: Congruence Classes of Simplex Structures in Finite Field Vector Spaces
Congruence Classes of Simplex Structures in Finite Field Vector Spaces Open
We study a generalization of the Erdős-Falconer distance problem over finite fields. For a graph $G$, two embeddings $p, p': V(G) \to \mathbb{F}_q^d$ of a graph $G$ are congruent if for all edges $(v_i, v_j)$ of $G$ we have that $||p(v_i) …
View article: Packing sets in Euclidean space by affine transformations
Packing sets in Euclidean space by affine transformations Open
For Borel subsets $Θ\subset O(d)\times \mathbb{R}^d$ (the set of all rigid motions) and $E\subset \mathbb{R}^d$, we define \begin{align*} Θ(E):=\bigcup_{(g,z)\in Θ}(gE+z). \end{align*} In this paper, we investigate the Lebesgue measure and…
View article: Multi-linear forms, structure of graphs and Lebesgue spaces
Multi-linear forms, structure of graphs and Lebesgue spaces Open
Consider the operator $$T_Kf(x)=\int_{{\mathbb R}^d} K(x,y) f(y) dy,$$ where $K$ is a locally integrable function or a measure. The purpose of this paper is to study the multi-linear form $$ Λ^K_G(f_1, \dots, f_n)=\int \dots \int \prod_{ \…
View article: A quantitative version of the Steinhaus theorem
A quantitative version of the Steinhaus theorem Open
The classical Steinhaus theorem (\cite{Steinhaus1920}) says that if $A \subset {\Bbb R}^d$ has positive Lebesgue measure than $A-A=\{x-y: x,y \in A\}$ contains an open ball. We obtain some quantitative lower bounds on the size of this ball…
View article: Realizing trees of configurations in thin sets
Realizing trees of configurations in thin sets Open
Let $ϕ(x,y)$ be a continuous function, smooth away from the diagonal, such that, for some $α>0$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^ϕf(x)=\int_{ϕ(x,y)=t} f(y) ψ(y) dσ_{x,t}(y) \end{equation} map …
View article: Uncertainty Principles, Restriction, Bourgain's $\Lambda_Q$ Theorem, and Signal Recovery
Uncertainty Principles, Restriction, Bourgain's $\Lambda_Q$ Theorem, and Signal Recovery Open
View article: A distinction between the paraboloid and the sphere in weighted restriction
A distinction between the paraboloid and the sphere in weighted restriction Open
For several weights based on lattice point constructions in $\mathbb{R}^d (d \geq 2)$, we prove that the sharp $L^2$ weighted restriction inequality for the sphere is very different than the corresponding result for the paraboloid. The pro…
View article: $L^p$ integrability of functions with Fourier support on a smooth space curve
$L^p$ integrability of functions with Fourier support on a smooth space curve Open
We prove that if $f\in L^p(\mathbb{R}^k)$ with $p<(k^2+k+2)/2$ satisfies that $\widehat{f}$ is supported on a small perturbation of the moment curve in $\mathbb{R}^k$, then $f$ is identically zero. This improves the more general result of …
View article: Uncertainty Principles on Finite Abelian Groups, Restriction Theory, and Applications to Sparse Signal Recovery
Uncertainty Principles on Finite Abelian Groups, Restriction Theory, and Applications to Sparse Signal Recovery Open
Let $G$ be a finite abelian group. Let $f: G \to {\mathbb C}$ be a signal (i.e. function). The classical uncertainty principle asserts that the product of the size of the support of $f$ and its Fourier transform $\hat f$, $\text{supp}(f)$ …
View article: Simplices in thin subsets of Euclidean spaces
Simplices in thin subsets of Euclidean spaces Open
Let $\De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k
View article: Generalized point configurations in ${\mathbb F}_q^d$
Generalized point configurations in ${\mathbb F}_q^d$ Open
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…
View article: Improved bounds for embedding certain configurations in subsets of vector spaces over finite fields
Improved bounds for embedding certain configurations in subsets of vector spaces over finite fields Open
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 …
View article: VC-Dimension of Hyperplanes over Finite Fields
VC-Dimension of Hyperplanes over Finite Fields Open
Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where $h_y$ is the …
View article: A singular variant of the Falconer distance problem
A singular variant of the Falconer distance problem Open
In this paper we study the following variant of the Falconer distance problem. Let $E$ be a compact subset of ${\mathbb{R}}^d$, $d \ge 1$, and define $$ \Box(E)=\left\{\sqrt{{|x-y|}^2+{|x-z|}^2}: x,y,z \in E,\, y\neq z \right\}.$$ We shall…
View article: An improved point-line incidence bound over arbitrary finite fields via the VC-dimension theory
An improved point-line incidence bound over arbitrary finite fields via the VC-dimension theory Open
The main purpose of this paper is to prove that the point-line incidence bound due to Vinh (2011) over arbitrary finite fields can be improved in certain ranges by using tools from the VC-dimension theory. As consequences, a number of appl…
View article: Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres Open
View article: The quotient set of the quadratic distance set over finite fields
The quotient set of the quadratic distance set over finite fields Open
Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\mathbb F_q$ and $E\subset \mathbb F_q^d$, we define $W(r)$ as the number of quadruples $(x,y,z,w)\in …