arXiv subjects

A. Iosevich

Publications and source records attributed to A. Iosevich.

At least 19 recordsLinked to original sources

arXiv2026

The Erd\H{o}s similarity conjecture and Rajchman measures

Let $A\subseteq\mathbb{R}$ support a probability measure whose Fourier--Stieltjes transform tends to zero at infinity. We prove that, for every $\varepsilon\in(0,1)$, there is a closed, $1$-periodic, nowhere dense set $E\subseteq\mathbb{R}$ such that \[ m(E\cap I)\ge1-\varepsilon \] for every interval $I$ of length $1$, while $E$ contains no affine copy of $A$. Thus every set supporting a Rajchman measure satisfies the Erd\H{o}s similarity conjecture in a uniform large-set form. The proof combines equidistribution modulo one for large dilates of the measure with a multiscale family of low-density periodic blockers; no quantitative rate of Fourier decay is used. We also refine a classical theorem of Iva\v{s}ev-Musatov, showing that for every Hausdorff gauge $h$ there is an $h$-null compact Rajchman support $K$ satisfying \[ \overline{\dim}_{\mathrm B}^{\log} K=\dim_{\mathrm P}^{\log} K=1. \] The value $1$ is sharp for both dimensions.

math.AP↗

arXiv2026

Elliptic curves, Fourier ratio, and sampling complexity

We study the normalized Frobenius trace associated with the Legendre family of elliptic curves over $\mathbb F_p$ from the point of view of Fourier complexity. If \[ f(t)=\frac{a_p(E_t)}{\sqrt p}, \qquad E_t:\ y^2=x(x-1)(x-t), \] with $f(0)=f(1)=0$, then \[ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}\asymp \sqrt p. \] More precisely, the Fourier transform of $f$ has squared $\ell^2$ norm of order $p$ while its individual coefficients remain uniformly bounded. It follows that no Fourier model supported on fewer than a sufficiently small constant multiple of $p$ frequencies can approximate $f$ in $\ell^2$ with error smaller than a fixed proportion of $\|f\|_2$. We also show that the Fourier magnitude profile of $f$ supports a family of at least $\exp(cp)$ real-valued functions with identical Fourier magnitudes and identical Fourier ratio, any two of which are separated by at least $c\sqrt p$ in $\ell^2$. Consequently, every deterministic reconstruction procedure that recovers all members of this family from bounded-precision point evaluations must use at least $c_Bp$ samples, where $c_B>0$ depends only on the number of bits used to encode each observation. The arithmetic input is unconditional and relies only on the Weil bound for mixed character sums, the evaluation of the quadratic Gauss sum, and elementary character identities.

math.NT↗

arXiv2026

The Erd\H{o}s Similarity Conjecture for Two-Fold Sumsets with a Geometric Summand

We settle a major case in the two-set regime of the Erd\H{o}s similarity conjecture: the sum of a geometric sequence and an arbitrary infinite set is never measure universal. Here a set $E\subset\R$ is measure universal if every measurable set of positive Lebesgue measure contains an affine copy of $E$. More precisely, if $A\subset\R$ is infinite, $a\neq 0$, and $0<|r|<1$, then neither \[ \{ar^n:n\ge 1\}+A \qquad\text{nor}\qquad \{ar^n:n\ge 1\}-A \] is measure universal. More generally, the same conclusion holds when the geometric sequence is replaced by any set containing a lacunary sequence $(b_n)$ with $-\log b_n=O(n)$. Bourgain proved non-universality for sums of three arbitrary infinite sets, whereas the two-set regime is one of the principal remaining cases. Crucially, our conclusion applies to $\{2^{-n}\}+A$ for every infinite $A$, even though the non-universality of $\{2^{-n}\}$ itself remains open. The arbitrary-summand theorem is the maximal lacunary-density endpoint of a general counting-function trade-off. If $S_1,S_2\subset\R$ contain lacunary subsequences and $I(W),J(W)$ count their terms that are at least $e^{-W}$, then $S_1+S_2$ and $S_1-S_2$ are not measure universal whenever \[ \limsup_{W\to\infty}\frac{I(W)J(W)}{W}=\infty. \] No scale-separation or relative-decay assumption is required. The proof combines a finite-grid implementation of Kolountzakis' criterion with a near-additive-energy estimate controlling the clustering of lacunary cross-sums. A packing-number variant replaces lacunarity on one factor by a quantitative metric-mass condition. In particular, for $\alpha_1,\alpha_2>0$, the stretched-exponential sumset \[ \{2^{-n^{\alpha_1}}\}+\{2^{-n^{\alpha_2}}\} \] is not measure universal whenever $1/\alpha_1+1/\alpha_2>1$; analogous conclusions hold for difference sets.

math.CA↗

arXiv2026

Uncertainty principles and singular potentials

We establish uncertainty principles on compact Riemannian manifolds without boundary in the setting of Laplace-Beltrami operators, including the case of real-valued singular potentials. We replace the classical homogeneity assumption by a quantitative spectral condition and obtain corresponding stability versions of uncertainty inequalities. In particular, we prove that \[ (1-\epsilon-\epsilon')^2 \leq \frac{|E|}{|M|}\cdot \# X_S \cdot \sup_{x\in E} \frac{A_S(x)}{\frac{\# X_S}{|M|}}, \] which recovers the classical bound in the homogeneous case, quantifies its deterioration in the presence of spectral inhomogeneity, and is shown to be sharp in general. In {\it dimension one}, we show that the homogeneity condition holds automatically, and we complement this rigidity by incorporating Fourier-ratio complexity bounds, yielding a quantitative relationship between spectral complexity and spatial support. In higher dimensions, we derive analogous results using pointwise Weyl laws and the eigenfunction restriction estimates on submanifolds.

math.CA↗

arXiv2026

Arithmetic functions and learning theory

We establish a connection between analytic number theory and computational learning theory by showing that the M\"obius function belongs to a class of functions that is statistically hard to learn from random samples. Let $\mu_R$ denote the restriction of the M\"obius function to the squarefree integers in $\{1,\dots,R\}$. Using a recent lower bound of Pandey and Radziwi{\l}{\l} for the $L^1$ norm of exponential sums with M\"obius coefficients, we prove that \[ \FR(\mu_R) \gg R^{-1/4-\epsilon} \] for every $\epsilon>0$. We then show that, for a suitable absolute constant $c_0>0$, the class of $\{-1,1\}$-valued functions on the squarefree integers with Fourier Ratio at least $c_0$ has Vapnik--Chervonenkis dimension at least $cR$. It follows that any distribution-independent learning algorithm that succeeds uniformly on the class $\mathcal{H}_R(\eta_R)$ containing $\mu_R$, where $\eta_R \to 0$, requires at least $\Omega(R)$ samples. We also discuss a conditional improvement under a strong uniform bound for additive twists of the M\"obius function, and we note that the same method applies to the Liouville function.

math.NT↗

arXiv2026

High-Dimensional Signal Compression: Lattice Point Bounds and Metric Entropy

We study worst-case signal compression under an $\ell^2$ energy constraint, with coordinate-dependent quantization precisions. The compression problem is reduced to counting lattice points in a diagonal ellipsoid. Under balanced precision profiles, we obtain explicit, dimension-dependent upper bounds on the logarithmic codebook size. The analysis refines Landau's classical lattice point estimates using uniform Bessel bounds due to Olenko and explicit Abel summation.

cs.IT↗

arXiv2026

Falconer lattice sets and the Erdos similarity problem

We show that a family of extremely thin sets satisfy the Erd\H{o}s similarity conjecture. These examples lie outside the range covered by recent work of Shmerkin and Yavicoli \cite{ShmerkinYavicoli2025}. As we shall see, they have small logarithmic dimension. They do not contain affine copies of slowly decaying sequences, so the result does not follow from earlier work of Falconer and Eigen \cite{Falconer1984,Eigen}. On the other hand, they do contain sequences of rapid decay, for which the conjecture is still open in general. Our argument is based on Falconer lattice sets and a theorem of Bourgain \cite{Bourgain2003}.

math.CA↗

arXiv2026

Spectral synthesis with the complexity parameter

We show that spectral synthesis thresholds are governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, we prove that if a compactly supported measure has finite $\alpha$-dimensional packing measure and the associated Fourier ratio decays with asymptotic exponent $\kappa$, then the classical synthesis threshold improves from $\frac{2d}{\alpha}$ to $\frac{2(d-2\kappa)}{\alpha-2\kappa}$. We then establish an analogous result on compact Riemannian manifolds without boundary. In that setting the relevant object is a localized spectral Fourier ratio defined using Laplace--Beltrami spectral projectors. The resulting synthesis threshold is again determined by the decay exponent of this complexity parameter. These results place Euclidean and manifold spectral synthesis into a common framework in which geometric size and spectral complexity jointly govern uniqueness

math.CA↗

arXiv2026

Spectral synthesis on Riemannian manifolds

We study spectral synthesis for measures supported on thin subsets of compact Riemannian manifolds. We prove that under natural non-concentration conditions, such measures admit quantitative spectral synthesis, with explicit stability bounds. We show that this phenomenon depends strongly on the underlying geometry. On the torus, synthesis holds under broad assumptions, while on the sphere we establish rigidity results demonstrating that synthesis can fail in a sharp sense. As consequences, we obtain quantitative approximation results and uncertainty principles for functions with thin spectral support. These results provide a unified framework connecting spectral synthesis, geometric structure, and stability on compact manifolds.

math.CA↗

arXiv2026

PDE propagation, sampling, and the Fourier ratio

We study recovery from incomplete random spatial samples for discretized fields arising as fixed-time snapshots of partial differential equations. The organizing parameter is the Fourier ratio $$ FR(g)=\frac{\|\widehat g\|_1}{\|\widehat g\|_2}, $$ which quantifies effective spectral dimension and governs stable $\ell^1$ recovery in bounded orthonormal sampling models. Our main observation is that fixed-time PDE propagation can strictly improve Fourier ratio bounds relative to the discretized initial data. In dimension three, the wave snapshot operator introduces additional high-frequency decay, leading after discretization to Fourier ratio bounds that are uniformly controlled in the grid size (up to discretization errors), whereas the corresponding bounds for the initial discretization are typically polynomial in $N$. For the heat equation in any dimension, Gaussian frequency damping yields Fourier ratio bounds that are essentially independent of grid resolution for fixed positive time. Combining these deterministic Fourier ratio improvements with standard $\ell^1$ recovery guarantees yields explicit sampling-rate bounds for stable reconstruction from missing spatial samples. Numerical experiments confirm that PDE propagation acts as a spectral preconditioner that lowers effective sampling complexity in practice.

math.NA↗

arXiv2026

Discretization, sampling, and the Fourier ratio

We derive fundamental sampling bounds for smooth signals in continuous settings without sparsity assumptions. By introducing the Fourier ratio as a measure of spectral compressibility induced by smoothness, we obtain explicit, deterministic bounds linking signal regularity to recoverability from incomplete random samples. For functions in $C^{2}([0,1]^{2})$ sampled on an $N$ by $N$ grid, we show that a random subset of spatial samples of size $$ C\frac{r_{N}^{2}}{\eps^{2}}\log(r_{N}/\eps)^{2}\log(N^{2}) $$ suffices, with high probability, to recover the entire discretized signal via $\ell^{1}$ minimization with relative $L^{2}$ error $O(\eps)$. We develop a parallel theory for bandlimited functions on the unit sphere, obtaining analogous recovery guarantees with sample complexity scaling polylogarithmically in the bandwidth. Our results establish smoothness as a deterministic prior that enforces compressibility in the Fourier domain, bridging continuous harmonic analysis with discrete compressed sensing in a unified information-theoretic framework.

math.CA↗

arXiv2026

A Refinement of Vapnik--Chervonenkis' Theorem

Vapnik--Chervonenkis' theorem is a seminal result in machine learning. It establishes sufficient conditions for empirical probabilities to converge to theoretical probabilities, uniformly over families of events. It also provides an estimate for the rate of such uniform convergence. We revisit the probabilistic component of the classical argument. Instead of applying Hoeffding's inequality at the final step, we use a normal approximation with explicit Berry--Esseen error control. This yields a moderate-deviation sharpening of the usual VC estimate, with an additional factor of order $(\varepsilon\sqrt{n})^{-1}$ in the leading exponential term when $\varepsilon\sqrt{n}$ is large.

cs.LG↗

arXiv2025

The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures

We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure $\mu$ on $\mathbb{R}^d$ and $f\in L^2(\mu)$, the Fourier ratio compares $L^1$ and $L^2$ norms of a regularized Fourier transform at scale $R$. We develop a fractal uncertainty principle giving sharp two-sided bounds in terms of covering numbers of spatial and frequency supports, with applications to exact signal recovery. We show that small Fourier ratio implies efficient approximation by low-degree trigonometric polynomials in $L^1$, $L^2$, and $L^\infty$. In contrast, restriction estimates reveal a sharp gap between curved measures and random fractal measures, yielding strong lower bounds on approximation degree. Applications to convex surface measures are also obtained.

math.CA↗

arXiv2025

The Fourier Ratio and complexity of signals

We study the Fourier ratio of a signal $f:\mathbb Z_N\to\mathbb C$, \[ \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{\|\widehat f\|_{L^1(\mu)}}{\|\widehat f\|_{L^2(\mu)}} \ =\ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}, \] as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small $\mathrm{FR}(f)$ forces $f$ to be well-approximated in both $L^2$ and $L^\infty$ by low-degree trigonometric polynomials. Quantitatively, the class $\{f:\mathrm{FR}(f)\le r\}$ admits degree $O(r^2)$ $L^2$-approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.

math.CA↗

arXiv2025

Orlicz spaces and the uncertainty principle

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, Iosevich and Mayeli improved the uncertainty principle for signals with Fourier supported on generic sets. This was done by employing the Fourier restriction theory in $L^p$ spaces. In this paper, we extended the $(p,q)$-restriction setting to Orlicz spaces. Then we apply uncertainty principles to the problem of exact recovery, which again extends and recovers the result that Iosevich and Mayeli obtained in Lebesgue spaces.

math.CA↗

arXiv2025

On spectral synthesis in ${\mathbb Z}_N^d$

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 $f \in L^p({\mathbb R}^d)$ for $p \leq \frac{2n}{d}$, then $f$ is identically equal to $0$. In this paper, we investigate an analogous problem for functions $f: {\mathbb Z}_N^d \to {\mathbb C}$. Bourgain's celebrated result on $\Lambda_p$ sets (\cite{Bou89}), random constructions (\cite{Bab89}), and connections with the theory of exact signal recovery (\cite{DS89}, \cite{MS73}, \cite{IKLM24}, \cite{IM24}) play an important role.

math.CA↗

arXiv2025

Additive energy, uncertainty principle and signal recovery mechanisms

Given a signal $f:G\to\mathbb{C}$, where $G$ is a finite abelian group, under what reasonable assumptions can we guarantee the exact recovery of $f$ from a proper subset of its Fourier coefficients? In 1989, Donoho and Stark established a result \cite{DS89} using the classical uncertainty principle, which states that $|\text{supp}(f)|\cdot|\text{supp}(\hat{f})|\geq |G|$ for any nonzero signal $f$. Another result, first proven by Santose and Symes \cite{SS86}, was based on the Logan phenomenon \cite{L65}. In particular, the result showcases how the $L^1$ and $L^2$ minimizing signals with matching Fourier frequencies often recovers the original signal. The purpose of this paper is to relate these recovery mechanisms to additive energy, a combinatorial measure denoted and defined by $$\Lambda(A)=\left| \left\{ (x_1, x_2, x_3, x_4) \in A^4 \mid x_1 + x_2 = x_3 + x_4 \right\} \right|,$$ where $A\subset\mathbb{Z}_N^d$. In the first part of this paper, we use combinatorial techniques to establish an improved variety of the uncertainty principle in terms of additive energy. In a similar fashion as the Donoho-Stark argument, we use this principle to establish an often stronger recovery condition. In the latter half of the paper, we invoke these combinatorial methods to demonstrate two $L^p$ minimizing recovery results.

math.CA↗

arXiv2024

Effective support, Dirac combs, and signal recovery

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 if a subset of frequencies ${\{\widehat{f}(m)\}}_{m \in S}$ of $f$ are unobserved due to noise or other interference, then $f$ can be recovered exactly and uniquely provided that $$ |E| \cdot |S|<\frac{N^d}{2},$$ where $E$ is the support of $f$, i.e., $E=\{x \in {\mathbb Z}_N^d: f(x) \not=0\}$. In this paper, we consider signals that are Dirac combs of complexity $\gamma$, meaning they have the form $f(x)=\sum_{i=1}^{\gamma} a_i 1_{A_i}(x)$, where the sets $A_i \subset {\mathbb Z}_N^d$ are disjoint, $a_i$ are complex numbers, and $\gamma \leq N^d$. We will define the concept of effective support of these signals and show that if $\gamma$ is not too large, a good recovery condition can be obtained by pigeonholing under additional reasonable assumptions on the distribution of values. Our approach produces a non-trivial uncertainty principle and a signal recovery condition in many situations when the support of the function is too large to apply the classical theory.

math.CA↗