arXiv subjects

Alex Iosevich

Publications and source records attributed to Alex Iosevich.

At least 19 recordsLinked to original sources

arXiv2026

Minkowski sums with convex curves without pointwise Fourier decay

Let $\Gamma\subset\mathbb R^2$ be a compact convex graph and define \[ T(\Gamma) = \inf \left\{ t: \dim_{\mathrm H}(E)>t \Longrightarrow |E+\Gamma|>0 \text{ for every compact }E\subset\mathbb R^2 \right\}. \] For a graph over an interval of positive length the smallest possible value is $T(\Gamma)=1$. We ask whether this optimal conclusion can hold when pointwise Fourier decay of arclength is unavailable. The answer is yes, even for strictly convex curves. We use the Fourier transform convention $\widehat\nu(\xi)=\int e^{-2\pi i x\cdot\xi}\,d\nu(x)$. We construct a strictly convex Lipschitz graph $\Gamma$ with $T(\Gamma)=1$ such that, for every nontrivial subarc $\Gamma_0$ and every $\alpha>0$, \[ \limsup_{|\xi|\to\infty} |\xi|^\alpha \left| \widehat{H^1|_{\Gamma_0}}(\xi) \right|= \infty. \] We also give a convex example for which arclength on every nontrivial subarc fails even to be a Rajchman measure. The geometric mechanism behind these examples is a positive curved trace: if $\Gamma$ contains a positive-length subset of a $C^2$ curve whose curvature is bounded away from zero, then $|E+\Gamma|>0$ whenever $\dim_{\mathrm H}(E)>1$. For a nondegenerate graph this gives $T(\Gamma)=1$. For convex graphs it implies, in particular, that $T(\Gamma)=1$ whenever the curvature measure has a nonzero absolutely continuous part. The positive-measure proofs are in physical space and use translated-tube intersections and elementary facts about convex functions. The same overlap estimates give Mattila-type lower bounds for the average lengths of the associated curve projections of neighborhoods under the positive curved-trace hypothesis. We also prove a dimension-one endpoint result for sets with a positive-length rectifiable part and formulate the main remaining question: whether every strictly convex Lipschitz graph has the optimal threshold $T(\Gamma)=1$.

math.CA↗

arXiv2026

Edge complexity of graphs

Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.

math.CO↗

arXiv2026

Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets

Let $d \geq 3$, \[ D \subsetneq \{ 0,1,\dots,d-1\}, \qquad |D| \geq 2, \ 0 \in D \] be a finite alphabet, and define the integer Cantor set \begin{align} \mathcal{C} := \mathcal{C}_{D} := \bigcup_{J \geq 0} \Big\{ \sum_{j =0}^J a_j d^j : a_j \in D \Big\}. \end{align} We prove that for any $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and any $f \in L^p(X)$, $2\leq p<\infty$, the ergodic averages \begin{align} \frac{1}{|\mathcal{C}_N|} \sum_{n \in \mathcal{C}_N } f(T^n x), \qquad \mathcal{C}_N := \mathcal{C} \cap \{1,2,\dots,N \} \end{align} converge $\mu$-almost everywhere. By rescaling, this allows us to resolve the question of lacunary differentiation of Cantor measures at self-similar scales: if \begin{align} \mathcal{C}' := \Big\{ \sum_{j \geq 1} a_j d^{-j} : a_j \in D \Big\} \subset [0,1] \end{align} is a real-variable Cantor set, and $\nu$ denotes its natural measure, then we prove that \begin{align} \lim_{k \to \infty} \int f(x-d^{-k} t) \ d\nu(t) = f(x) \end{align} Lebesgue almost-everywhere for any $f \in L^2_{\text{loc}}(\mathbb{R})$.

math.DS↗

arXiv2026

Metric entropy of Fourier ratio classes on ${\mathbb Z}_N$

We study metric entropy and uniform sampling for classes of signals on ${\mathbb Z}_N$ with prescribed Fourier ratio. The Fourier ratio measures how spread out the Fourier transform of a signal is, interpolating between sparse spectral support and nearly uniform spectral distribution. Our main result gives upper and lower bounds for the metric entropy of a Fourier-ratio layer of size $r.$ At any sufficiently small fixed covering scale, these bounds match in their dependence on $r$ and $N$ and show that $FR(f)^2$ acts as an effective dimension parameter governing the size of the class. We use the entropy estimate to obtain uniform bounds for empirical approximation over Fourier-ratio classes. We also establish a phase-orbit packing result. If a single signal has a flat spectral block of size $k,$ then phase perturbations of that signal generate an exponentially large family with the same Fourier ratio and positive $\ell^2$ separation. Together, these results show that the Fourier ratio governs not only approximation properties of individual signals, but also the geometric size and uniform sampling behavior of entire signal classes.

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arXiv2026

Fourier Ratios of Graph Kernels: Energy Bounds, Optimal Labelings, and Recovery

We study labeling-sensitive Fourier complexity for finite graph kernels. After identifying the vertices of a graph with the cyclic group $\mathbb Z_N$, its adjacency matrix becomes a function on $\mathbb Z_N^2$. Minimizing the quotient of the $\ell^1$ and $\ell^2$ norms of its two-dimensional Fourier transform over all vertex labelings gives an isomorphism invariant $\operatorname{FR}_{\min}(G)$. A nuclear-norm argument gives \[\operatorname{FR}_{\min}(G) \geq \frac{\mathcal E(G)}{\sqrt{2s}},\] where $s$ is the number of edges and $\mathcal E(G)$ is the graph energy. The natural cyclic labeling attains equality for every circulant graph. We obtain exact formulas for several graph families and a labeling-sensitive complete bipartite example. We also connect the invariant with the Fourier algebra of $\mathbb Z_N^2$. The quantitative Cohen idempotent theorem implies that every Boolean kernel of bounded Fourier ratio has an exact signed coset decomposition whose length is independent of $N$. A previously established Fourier-ratio recovery theorem gives stable Frobenius approximation of a fixed labeled adjacency matrix from Bernoulli samples. We distinguish this conclusion from exact edge recovery and from the problem of finding a good labeling. For a Laplacian eigenvalue of multiplicity $m(\lambda)$, we prove \[ \operatorname{FR}_{\min}(\Pi_\lambda) \geq \sqrt{m(\lambda)}, \] with equality for circulant graphs. Strongly regular graphs and the Petersen graph show how adjacency and projector complexity can agree or differ. Direct projector sampling yields heat-kernel approximation. We conclude with a graph-signal spectral synthesis principle and asymptotic uniqueness from incomplete vertex data.

math.CO↗

arXiv2026

Positive Measure of Unions of Variable Surfaces

Let $E \subset \mathbb R^d$, $d \ge 2$, be compact, and let $\phi(x,y)$ be a smooth function satisfying the Phong--Stein rotational curvature condition on $\{\phi(x,y)=1\}$. We prove that if $\dim_{\mathcal H}(E)>1$, then $$ \left|\bigcup_{x \in E} \{y : \phi(x,y)=1\}\right|>0. $$ This extends the positivity theorem of Mitsis ($d\geq3$) and Wolff ($d=2$) for spheres to a general variable coefficient setting via $L^2$ estimates for Fourier integral operators. The argument also shows that positivity is stable under finite-order degeneracies of the Monge--Amp\`ere determinant through the weighted averaging theory of Sogge and Stein. We next consider variable level sets $$ \Sigma_x=\{y:\phi(x,y)=t(x)\}, $$ where $t(x)$ is measurable. A maximal operator argument yields positivity under the condition $\dim_{\mathcal H}(E)>2$. We show that this loss reflects a genuine geometric obstruction related to Kakeya-type compression phenomena. In contrast, under a direct geometric intersection hypothesis controlling overlaps of the hypersurfaces $\Sigma_x$, we recover the full threshold $\dim_{\mathcal H}(E)>1$ for arbitrary measurable selections $t=t(x)$. At the endpoint $\dim_{\mathcal H}(E)=1$, we obtain positivity under the additional assumption that $E$ is $1$-rectifiable with $\mathcal H^1(E)>0$. We also show that positivity of Lebesgue measure does not in general imply interior regularity: even for large or rectifiable parameter sets, the resulting unions may have empty interior. Finally, we discuss extensions to higher co-dimension families and the role of geometric structure in preventing compression phenomena.

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arXiv2026

Uncertainty Principles, Spectral Localization, and Singular Schr\"odinger Operators on Compact Manifolds

We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schr\"odinger operators. Our results relate the size of the support of spectrally localized functions to the cardinality of the underlying spectral cluster and to Fourier-ratio type quantities. We obtain analogues for Schr\"odinger operators with singular potentials belonging to Kato and scaling-critical classes. As an application, we prove uniqueness results for recovery from incomplete spectral data on compact manifolds. Under curvature assumptions, including nonpositive and negative sectional curvatures, we also prove logarithmically improved uncertainty principles associated with shrinking spectral windows.

math.AP↗

arXiv2026

Large values in time series and additive combinatorics

It is well-known in industrial data science that large values of real-life time series tend to be structured and often follow concrete and visible patterns. In this paper, we use ideas from additive combinatorics and discrete Fourier analysis to give this heuristic a mathematical foundation. Our main tool is the Fourier ratio, a complexity measure previously used in compressed sensing, combined with a generalized version of Chang's lemma from additive combinatorics. Together, these yield a precise prediction: when the Fourier ratio of a time series is small, the set of its largest values can be additively generated by a very small set using only $\{-1,0,1\}$ coefficients. We test this prediction on US inflation data and Delhi climate data, both in their original form and after mean-centering. The numerical results confirm the predicted structure: a generating set of size $4$--$7$ suffices to span large spectra containing dozens of points, even when the Fourier ratio is large enough that our theoretical bounds become loose. These findings provide a rigorous explanation for why extreme values in real-world data are information-rich and structurally significant.

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arXiv2026

The Fourier Ratio: A Unifying Measure of Complexity for Recovery, Localization, and Learning

We introduce a generalized Fourier ratio, the $\ell^1/\ell^2$ norm ratio of coefficients in an \emph{arbitrary} orthonormal system, as a single, basis-invariant measure of \emph{effective dimension} that governs fundamental limits across signal recovery, localization, and learning. First, we prove that functions with small Fourier ratio can be stably recovered from random missing samples via $\ell^1$ minimization, extending and clarifying compressed sensing guarantees for general bounded orthonormal systems. Second, we establish a sharp \emph{localization obstruction}: any attempt to localize recovery to subslices of a product space necessarily inflates the Fourier ratio by a factor scaling with the square root of the slice count, demonstrating that global complexity cannot be distributed locally. Finally, we show that the same parameter controls key complexity-theoretic measures: it provides explicit upper bounds on Kolmogorov rate-distortion description length and on the statistical query (SQ) dimension of the associated function class. These results unify analytic, algorithmic, and learning-theoretic constraints under a single complexity parameter, revealing the Fourier ratio as a fundamental invariant in information-theoretic signal processing.

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arXiv2025

Signal recovery using Gabor frames

We present a novel probabilistic framework for the recovery of discrete signals with missing data, extending classical Fourier-based methods. While prior results, such as those of Donoho and Stark; see also Logan's method, guarantee exact recovery under strict deterministic sparsity constraints, they do not account for stochastic patterns of data loss. Our approach combines a row-wise Gabor transform with a probabilistic model for missing frequencies, establishing near-certain recovery when losses occur randomly. The key innovation is a maximal row-support criterion that allows unique reconstruction with high probability, even when the overall signal support significantly exceeds classical bounds. Specifically, we show that if missing frequencies are independently distributed according to a binomial law, the probability of exact recovery converges to $1$ as the signal size grows. This provides, to our knowledge, the first rigorous probabilistic recovery guarantee exploiting row-wise signal structure. Our framework offers new insights into the interplay between sparsity, transform structure, and stochastic loss, with immediate implications for communications, imaging, and data compression. It also opens avenues for future research, including extensions to higher-dimensional signals, adaptive transforms, and more general probabilistic loss models, potentially enabling even more robust recovery guarantees.

math.CA↗

arXiv2025

Refined additive uncertainty principle

Signal recovery from incomplete or partial frequency information is a fundamental problem in harmonic analysis and applied mathematics, with wide-ranging applications in communications, imaging, and data science. Historically, the classical uncertainty principles, such as those by Donoho and Stark, have provided essential bounds relating the sparsity of a signal and its Fourier transform, ensuring unique recovery under certain support size constraints. Recent advances have incorporated additive combinatorial notions, notably additive energy, to refine these uncertainty principles and capture deeper structural properties of signal supports. Building upon this line of work, we present a strengthened additive energy uncertainty principle for functions $f:\mathbb{Z}_N^d\to\mathbb{C}$, introducing explicit correction terms that measure how far the supports are from highly structured extremal sets like subgroup cosets. We have two main results. Our first theorem introduces a correction term which strictly improves the additive energy uncertainty principle from Aldahleh et al., provided that the classical uncertainty principle is not satisfied with equality. Our second theorem uses the improvement to obtain a better recovery condition. These theorems deliver strictly improved bounds over prior results whenever the product of the support sizes differs from the ambient dimension, offering a more nuanced understanding of the interplay between additive structure and Fourier sparsity. Importantly, we leverage these improvements to establish sharper sufficient conditions for unique and exact recovery of signals from partially observed frequencies, explicitly quantifying the role of additive energy in recoverability.

math.CA↗

arXiv2025

The VC-dimension and point configurations in $\mathbb{R}^d$

Given a set $X$ and a collection ${\mathcal H}$ of functions from $X$ to $\{0,1\}$, the VC-dimension measures the complexity of the hypothesis class $\mathcal{H}$ in the context of PAC learning. In recent years, this has been connected to geometric configuration problems in vector spaces over finite fields. In particular, it is easy to show that the VC-dimension of the set of spheres of a given radius in $\mathbb{F}_q^d$ is equal to $d+1$, since this is how many points generically determine a sphere. It is known that for $E\subseteq \mathbb{F}_q^d$, $|E|\geq q^{d-\frac{1}{d-1}}$, the set of spheres centered at points in $E$, and intersected with the set $E$, has VC-dimension either $d$ or $d+1$. In this paper, we study a similar question over Euclidean space. We find an explicit dimensional threshold $s_d<d$ so that whenever $E\subseteq \mathbb{R}^d$, $d\geq 3$, and the Hausdorff dimension of $E$ is at least $s_d$, it follows that there exists an interval $I$ such that for any $t\in I$, the VC-dimension of the set of spheres of radius $t$ centered at points in $E$, and intersected with $E$, is at least $3$. In the process of proving this theorem, we also provide the first explicit dimensional threshold for a set $E\subseteq \mathbb{R}^3$ to contain a $4$-cycle, i.e. $x_1,x_2,x_3,x_4\in E$ satisfying $$ |x_1-x_2|=|x_2-x_3|=|x_3-x_4|=|x_4-x_1| $$

math.CA↗

arXiv2025

Entropy Expansion for General Polynomial Images of Frostman Random Variables

We prove dyadic entropy expansion for the observables $X+Y$ and $f(X,Y)$ under Frostman nonconcentration hypotheses on a prescribed, possibly dependent law. For an integer $n\ge1$, write $H_n(Z)=H(\lfloor 2^nZ\rfloor)$ for the base-two Shannon entropy at resolution $2^{-n}$; thus $n$ indexes the fineness of the dyadic discretization. For every $0 4/3$, we obtain $\max\{H_n(X+Y),H_n(f(X,Y))\}\ge (\frac{s_1+s_2}{2}+\varepsilon)n-O(1)$ for an explicit $\varepsilon>0$. Here the baseline averages the two Frostman exponents. We also classify the exceptional coordinate representations and construct obstructions for algebraic affine directions with Frostman constants uniform in the scale.

math.CA↗

arXiv2025

Fourier minimization and imputation of time series

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, and those values need to be filled in before the data set can be properly analyzed. Many freely available forecasting software packages, such as the sktime library, have built-in mechanisms for filling in missing values. The purpose of this paper is to adapt the classical $L^1$ minimization method for signal recovery to the filling of missing values in times. The theoretical justifications of these methods leverage results by Bourgain (\cite{Bourgain89}), Talagrand (\cite{Talagrand98}), the second and the third listed authors (\cite{IM24}), and the result by the second listed author, Kashin, Limonova and the third listed author (\cite{IKLM24}). Brief numerical tests for these algorithms are given but more extensive will be discussed in a companion paper.

math.CA↗

arXiv2025

An Approach To Endpoint Problems in Oscillatory Singular Integrals

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 and stationary phase considerations.

math.CA↗

arXiv2024

Fourier Uncertainty Principles on Riemannian Manifolds

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 of eigenfunctions, and connections to Bourgain's celebrated $\Lambda_q$ theorem are discussed in this context.

math.CA↗

arXiv2024

Packing sets under finite groups via algebraic incidence structures

Let $G$ be a finite group acting on a vector space $V = \mathbb{F}_p^n$ over a prime field. Given finite sets $S \subset G$ and $E \subset V$, we study the restricted orbit union $S(E) = \bigcup_{g\in S} g(E)$ and establish quantitative lower bounds for $|S(E)|$ in terms of $|S|$, $|E|$, and natural structural conditions. This finite field packing problem has connections to distance geometry, configuration counting, and expanding graphs. For $G = SL_2(\mathbb{F}_p)$ acting on $\mathbb{F}_p^2$, we prove that $$|S(E)| \gg \min\left\lbrace p^2, \frac{|S||E|}{p^2}\right\rbrace,$$ which is sharp. Under geometric non-concentration conditions on $E$ and subgroup-avoidance hypotheses on $S$, we obtain a power-saving improvement of the form $$|S(E)|\gg \min \left\lbrace p^2, ~\max\left\lbrace\frac{|S||E|}{pk}, ~\frac{|S|^{\frac{1}{2}}|E|}{p^{\frac{1-\epsilon}{2}}k^{\frac{1}{2}}}\right\rbrace \right\rbrace,$$ where $k$ bounds the radial multiplicity of $E$. For small sets $|E| \leq p$, we establish optimal bounds using weighted incidence theory. Analogous results are proved for the first Heisenberg group $\mathbb{H}_1(\mathbb{F}_p)$ acting on $\mathbb{F}_p^3$. Our approach reformulates the problem as an incidence question in a bipartite action graph. The proofs combine Fourier analytic techniques, energy estimates, point-line incidence bounds, and area-energy inequalities for skew dot products. The methods extend classical sum-product type problems and incidence theory to noncommutative group actions.

math.CO↗

arXiv2024

Congruence Classes of Simplex Structures in Finite Field Vector Spaces

We study a generalization of the Erd\H{o}s-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 $||p(v_i) - p(v_j)|| = ||p'(v_i) - p'(v_j)||$. What is the infimum of $s$ such that for any subset $E\subset \mathbb{F}_q^d$ with $|E| \gtrsim q^s$, $E$ contains a positive proportion of congruence classes of $G$ in $\mathbb{F}_q^d$? Bennett et al. and McDonald used group action methods to prove results in the case of $k$-simplices. The work of Iosevich, Jardine, and McDonald as well as that of Bright et al. have proved results in the case of trees and trees of simplices, utilizing the inductive nature of these graphs. Recently, Aksoy, Iosevich, and McDonald combined these two approaches to obtain nontrivial bounds on the "bowtie" graph, two triangles joined at a vertex. Their proof relies on an application of the Hadamard three-lines theorem to pass to a different graph. We develop novel geometric techniques called branch shifting and simplex unbalancing to reduce our analysis of trees of simplices to a much smaller class of simplex structures. This allows us to establish a framework that handles a wide class of graphs exhibiting a combination of rigid and loose behavior. In $\mathbb{F}_q^2$, this approach gives new nontrivial bounds on chains and trees of simplices. In $\mathbb{F}_q^d$, we improve on the results of Bright et al. in many cases and generalize their work to a wider class of simplex trees. We discuss partial progress on how this framework can be extended to more general simplex structures, such as cycles of simplices and structures of simplices glued together along an edge or a face.

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