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Alex Iosevich

Publications and source records attributed to Alex Iosevich.

At least 55 records · Page 3Linked to original sources
arXiv2023

Multi-linear forms, graphs, and $L^p$-improving measures in ${\Bbb F}_q^d$

The purpose of this paper is to introduce and study the following graph theoretic paradigm. Let $$T_Kf(x)=\int K(x,y) f(y) dμ(y),$$ where $f: X \to {\Bbb R}$, $X$ a set, finite or infinite, and $K$ and $μ$ denote a suitable kernel and a measure, respectively. Given a connected ordered graph $G$ on $n$ vertices, consider the multi-linear form $$ Λ_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) dμ(x^l), $$ where ${\mathcal E}(G)$ is the edge set of $G$. Define $Λ_G(p_1, \ldots, p_n)$ as the smallest constant $C>0$ such that the inequality $$ Λ_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, μ)}$$ holds for all non-negative real-valued functions $f_i$, $1\le i\le n$, on $X$. The basic question is, how does the structure of $G$ and the mapping properties of the operator $T_K$ influence the sharp exponents. In this paper, this question is investigated mainly in the case $X={\Bbb F}_q^d$, the $d$-dimensional vector space over the field with $q$ elements, and $K(x^i,x^j)$ is the indicator function of the sphere evaluated at $x^i-x^j$. This provides a connection with the study of $L^p$-improving measures and distance set problems.

math.CA↗
arXiv2023

Approximate orthogonality, Bourgain's pinned distance theorem and exponential frames

Let $A$ be a countable and discrete subset of ${\Bbb R}^d$, $d \ge 2$, of positive upper Beurling density. Let $K$ denote a bounded symmetric convex set with a smooth boundary and everywhere non-vanishing Gaussian curvature. It is known that ${\mathcal E}(A)=\{e^{2 πi x \cdot a}\}_{a \in A}$ cannot serve as an orthogonal basis for $L^2(K)$ \cite{IKT01}. In this paper, we prove that even approximate average orthogonality is an obstacle to the existence of an exponential frame in the following sense. Let $A$ be as above and $ϕ\ge 0$ be a continuous monotonically nonincreasing function on $[0, \infty)$ such that the approximate orthogonality condition holds \begin{align}\notag {\left( \frac{1}{2^j} \int_{2^j}^{2^{j+1}} ϕ^p(t) dt \right)}^{1/p} \leq c_j 2^{-j\frac{d+1}{2}} \quad \text{and} \quad |\widehatχ_K(a-a')| \leq ϕ(ρ^*(a-a')) \ \forall a \not=a , a,a' \in A, \end{align} where $ρ^*$ is the Minkowski functional on $K^*$, the dual body of $K$. Then, if $$\limsup_{j \to \infty} c_j=0,$$ then the upper density of $A$ is equal to $0$, hence ${\mathcal E}(A)$ is not a frame for $L^2(K)$. The case $p=\infty$ was previously established by the authors of this paper in \cite{IM2020}. The point is that if ${\mathcal E}(A)$ is a frame for $L^2(K)$, then very few pairs of distinct exponentials $e^{2 πi x.a}, e^{2 πi x.a'}$ from ${\mathcal E}(A)$ come anywhere near being orthogonal. Our proof uses a generalization of Bourgain's result on pinned distances determined by sets of positive Lebesgue upper density in ${\Bbb R}^d$, $d \ge 2$. We also improve the $L^{\infty}$ version of this result originally established in \cite{IM2020}. By using an extension of the combinatorial idea from \cite{IR03}, we prove that under the $L^{\infty}$ hypothesis, $A$ is finite if $d \not=1 \mod 4$. If $d=1$ mod $4, A$ may be infinite, but if it is, then it must be a subset of a line.

math.CA↗
arXiv2023

Microlocal decoupling inequalities and the distance problem on Riemannian manifolds

We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran-Hickman-Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.

math.CA↗
arXiv2022

Pinned simplices and connections to product of sets on paraboloids

In this paper we obtain improved dimensional thresholds for dot product sets corresponding to compact subsets of a paraboloid. As a direct application of these estimates, we obtain significant improvements to the best known dimensional thresholds that guarantee that a given compact subset of Euclidean space determines a positive proportion of all possible congruence classes of simplexes. In many regimes this improves the results previously obtained by Erdogan-Hart-Iosevich (\cite{EHI}), Greenleaf-Iosevich-Liu-Palsson (\cite{GILP}) and others.

math.CA↗
arXiv2021

A multi-linear geometric estimate

We give a generalization of the geometric estimate used by Hart and the second author in their 2008 work on sums and products in finite fields. Their result concerned level sets of non-degenerate bilinear forms over finite fields, while in this work we prove that if $E\subset\mathbb{F}_q^d$ is sufficiently large and $\varpi$ is a non-degenerate multi-linear form then $\varpi$ will attain all possible nonzero values as its arguments vary over $E$, under a certain quantitative assumption on the extent to which $E$ is projective. We show that our bound is nontrivial in the case that $n=3$ and $d=2$ and construct examples of sets to which this applies. In particular, we give conditions under which every member of $\mathbb{F}_q^*$ belongs to $A\cdot A\cdot A+A\cdot A\cdot A\cdot A\cdot A\cdot A$ where $A$ is a union of cosets of a subgroup of $\mathbb{F}_q^*$.

math.NT↗
arXiv2021

Simplex Averaging Operators: Quasi-Banach and $L^p$-Improving Bounds in Lower Dimensions

We establish some new $L^p$-improving bounds for the $k$-simplex averaging operators $S^k$ that hold in dimensions $d \geq k$. As a consequence of these $L^p$-improving bounds we obtain nontrivial bounds $S^k\colon L^{p_1}\times\cdots\times L^{p_k}\rightarrow L^r$ with $r < 1$. In particular we show that the triangle averaging operator $S^2$ maps $ L^{\frac{d+1}{d}}\times L^{\frac{d+1}{d}} \rightarrow L^{\frac{d+1}{2d}}$ in dimensions $d\geq 2$. This improves quasi-Banach bounds obtained by Palsson and Sovine and extends bounds obtained by Greenleaf, Iosevich, Krauss, and Liu for the case of $k = d = 2$.

math.CA↗
arXiv2021

Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres

We establish an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{h_1, h_2, h_3}(λ): =\{x\in\mathbb Z_+^3:\lfloor h_1(x_1)\rfloor+\lfloor h_2(x_2)\rfloor+\lfloor h_3(x_3)\rfloor=λ\} \quad \text{with}\quad λ\in\mathbb Z_+; \] where functions $h_1, h_2, h_3$ are constant multiples of regularly varying functions of the form $h(x):=x^c\ell_h(x)$, where the exponent $c>1$ (but close to $1$) and a function $\ell_h(x)$ is taken from a certain wide class of slowly varying functions. Taking $h_1(x)=h_2(x)=h_3(x)=x^c$ we will also derive an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{c}^3(λ) := \{x \in \mathbb Z^3 : \lfloor |x_1|^c \rfloor + \lfloor |x_2|^c \rfloor + \lfloor |x_3|^c \rfloor= λ\} \quad \text{with}\quad λ\in\mathbb Z_+; \] which can be thought of as a perturbation of the classical Waring problem in three variables. We will use the latter asymptotic formula to study, the main results of this paper, norm and pointwise convergence of the ergodic averages \[ \frac{1}{\#\mathbf S_{c}^3(λ)}\sum_{n\in \mathbf S_{c}^3(λ)}f(T_1^{n_1}T_2^{n_2}T_3^{n_3}x) \quad \text{as}\quad λ\to\infty; \] where $T_1, T_2, T_3:X\to X$ are commuting invertible and measure-preserving transformations of a $σ$-finite measure space $(X, ν)$ for any function $f\in L^p(X)$ with $p>\frac{11-4c}{11-7c}$. Finally, we will study the equidistribution problem corresponding to the spheres $\mathbf S_{c}^3(λ)$.

math.DS↗
arXiv2021

On $k$-point configuration sets with nonempty interior

We give conditions for $k$-point configuration sets of thin sets to have nonempty interior, applicable to a wide variety of configurations. This is a continuation of our earlier work \cite{GIT19} on 2-point configurations, extending a theorem of Mattila and Sjölin \cite{MS99} for distance sets in Euclidean spaces. We show that for a general class of $k$-point configurations, the configuration set of a $k$-tuple of sets, $E_1,\,\dots,\, E_k$, has nonempty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizing $L^2$-Sobolev estimates of associated generalized Radon transforms over all nontrivial partitions of the $k$ points into two subsets. We illustrate the general theorems with numerous specific examples. Applications to 3-point configurations include areas of triangles in $\mathbb R^2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in $\mathbb R^3$; and ratios of pinned distances in $\mathbb R^2$ and $\mathbb R^3$. Results for 4-point configurations include cross-ratios on $\mathbb R$, triangle area pairs determined by quadrilaterals in $\mathbb R^2$, and dot products of differences in $\mathbb R^d$.

math.CA↗
arXiv2021

An improved result for Falconer's distance set problem in even dimensions

We show that if compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}$, where $d\geq 4$ is an even integer, then the distance set of $E$ has positive Lebesgue measure. This improves the previously best known result towards Falconer's distance set conjecture in even dimensions.

math.CA↗
arXiv2021

Existence of similar point configurations in thin subsets of $\Bbb R^d$

We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff measure in Euclidean space. These results can be viewed as variants, for thin sets, of theorems for sets of positive density in $\Bbb R^d$ due to Furstenberg, Katznelson and Weiss \cite{FKW90}, Bourgain \cite{B86} and Ziegler \cite{Z06}. Let $d \ge 2$ and $E\subset {\Bbb R}^d$ be a compact set. For $k\ge 1$, define $$Δ_k(E)=\left\{\left(|x^1-x^2|, \dots, |x^i-x^j|,\dots, |x^k-x^{k+1}|\right): \left\{x^i\right\}_{i=1}^{k+1}\subset E\right\} \subset {\Bbb R}^{k(k+1)/2}, $$ the {\it $(k+1)$-point configuration set} of $E$. For $k\le d$, this is (up to permutations) the set of congruences of $(k+1)$-point configurations in $E$; for $k>d$, it is the edge-length set of $(k+1)$-graphs whose vertices are in $E$. Previous works by a number of authors have found values $s_{k,d} s_{k,d}$, then $Δ_k(E)$ has positive Lebesgue measure. In this paper we study more refined properties of $Δ_k(E)$, namely the existence of (exactly) similar or multi--similar configurations. For $r\in\Bbb R,\, r>0$, let $$Δ_{k}^{r}(E):=\left\{\vec{t}\in Δ_k\left(E\right): r\vec{t}\in Δ_k\left(E\right)\right\}\subset Δ_k\left(E\right).$$ We show that for all $E$ with Hausdorff dimension $>s_{k,d}$, a natural measure $ν_k$ on $Δ_k(E)$ and all $r\in\Bbb R_+$, one has $ν_k\left(Δ_{k}^{r}\left(E\right)\right)>0$. Thus, there exist many pairs, $\{x^1, x^2, \dots, x^{k+1}\}$ and $\{y^1, y^2, \dots, y^{k+1}\}$, in $E$ which are similar by the scaling factor $r$. We also show the existence of triply-similar and multi-similar configurations.

math.CA↗
arXiv2021

Cycles of arbitrary length in distance graphs on $\mathbb{F}_q^d$

For $E \subset {\Bbb F}_q^d$, $d \ge 2$, where ${\Bbb F}_q$ is the finite field with $q$ elements, we consider the distance graph ${\mathcal G}^{dist}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $||x-y|| \equiv {(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=t$. We prove that if $|E| \ge C_k q^{\frac{d+2}{2}}$, then ${\mathcal G}^{dist}_t(E)$ contains a statistically correct number of cycles of length $k$. We are also going to consider the dot-product graph ${\mathcal G}^{prod}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $x \cdot y \equiv x_1y_1+\dots+x_dy_d=t$. We obtain similar results in this case using more sophisticated methods necessitated by the fact that the function $x \cdot y$ is not translation invariant. The exponent $\frac{d+2}{2}$ is improved for sufficiently long cycles.

math.CO↗
arXiv2020

Sums of Powers in Large Finite Fields: A Mix of Methods

Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal equations. Then, we offer two proofs, one new and elementary, and the other more classical, based on Fourier analysis and an application of a nontrivial estimate from the theory of finite fields. In context and juxtaposition, each will have its merits.

math.NT↗
arXiv2020

Simplices in thin subsets of Euclidean spaces

Let $\De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k<k$ such that any set $A\subs \R^k$ of Hausdorff dimension $dim\,A\geq s_k$ necessarily contains a similar copy of the simplex $\De$.

math.CA↗
arXiv2020

On complete and incomplete exponential systems

Given a bounded domain $Ω\subset {\Bbb R}^d$ with positive measure and a finite set $A=\{a^1, a^2, \dots, a^d\}$, we say that the set ${\mathcal E}(A)={\{e^{2 πi x \cdot a^j}\}}_{a^j \in A}$ is a complete exponential system if for every $ξ\in {\Bbb R}^d$, there exists $1 \leq j \leq d+1$ such that \begin{equation} \label{completedef} \int_Ω e^{-2 πi x \cdot (a^j-ξ)} dx \not=0; \end{equation} otherwise ${\mathcal E}(A)$ is called an incomplete exponential system. In this paper, we essentially classify complete and incomplete exponential systems when $Ω=B_d$, the unit ball, and when $Ω=Q_d$, the unit cube. Given a bounded domain $Ω$, we say that $e^{2 πi x \cdot a}, e^{2 πi x \cdot a'}$ are $ϕ$-approximately orthogonal if $$|\widehatχ_Ω(a-a')| \leq ϕ(|a-a'|), \ a\neq a'$$ where $ϕ: [0, \infty) \to [0, \infty)$ is a bounded measurable function that tends to $0$ at infinity. We prove that $L^2(B_d)$ does not possess a $ϕ$-approximate orthogonal basis of exponentials for a wide range of functions $ϕ$. The proof involves connections with the theory of distances in sets of positive Lebesgue upper density originally developed by Furstenberg, Katznelson and Weiss (\cite{FKW90}).

math.CA↗
arXiv2020

Falconer-type estimates for dot products

We present a family of sharpness examples for Falconer-type single dot product results. In particular, for $d\geq 2,$ for any $s<\frac{d+1}{2},$ we construct a Borel probability measure $μ$ satisfying the energy estimate $I_s(μ)<\infty,$ yet the estimate \begin{equation} (μ\times μ)\{(x,y):1\leq x\cdot y \leq 1+ε\} \leq Cε\end{equation} does not hold with constants independent of $ε$. It is known (\cite{EIT11}) that such an estimate always holds with $C$ independent of $ε$ if $I_{\frac{d+1}{2}}(μ)<\infty$. Thus our estimate proves the sharpness of the dimensional threshold in this result and generalizes similar results (\cite{Mat95}, \cite{IS16}) established in the case when the dot product $x \cdot y$ is replaced by the Euclidean distance function $|x-y|$, or, more generally, ${||x-y||}_K$, the distance that comes from the norm induced by a symmetric convex body $K$ with a smooth boundary and non-vanishing curvature. Our constructions are partially based on ideas that come from discrete incidence theory.

math.CA↗
arXiv2020

On restriction estimates for the zero radius sphere over finite fields

In this paper, we solve completely the $L^2\to L^r$ extension conjecture for the zero radius sphere over finite fields. We also obtain the sharp $L^p\to L^4$ extension estimate for non-zero radii spheres over finite fields, which improves significantly the previous result of the first and second listed authors.

math.CA↗
arXiv2019

An improved dimensional threshold for the angle problem

The Falconer distinct distance problem asks for a compact set $E\subset\mathbb{R}^d$ how large its Hausdorff dimension needs to be to ensure that the Lebesgue measure of its distance set is positive. In this paper we consider the analogous question for the set of angles. We show that if the Hausdorff dimension of $E$ is strictly bigger than $\frac{d}{2}$ then the Lebesgue measure of the angles set is positive. In the plane this result was previously established by Harangi et al. In higher dimensions, our exponent improves the $\frac{d+1}{2}$ threshold previously obtain by the authors of this paper and Mihalis Mourgoglou. We do not know what the right dimensional threshold should be in higher dimensions.

math.CA↗