arXiv subjects

A. Yavicoli

Publications and source records attributed to A. Yavicoli.

6 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

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

Falconer lattice sets and the Erdos similarity problem

We show that a family of extremely thin sets satisfy the Erdős 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

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↗

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 $μ$ on $\mathbb{R}^d$ and $f\in L^2(μ)$, 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↗