arXiv subjects

João P. G. Ramos

Publications and source records attributed to João P. G. Ramos.

At least 19 recordsLinked to original sources

arXiv2026

Phase retrieval for Schrödinger evolutions

Consider the one-dimensional Schrödinger evolution $$ i\partial_tw+\partial_x^2w=Vw,\hspace{5mm}(t,x)\in [0,T]\times\mathbb{R},$$ where $V$ is a real-valued time-dependent potential belonging to $L^1_tL^\infty_x+L^2_{t,x}$. We prove that an $L^2$ solution to the above equation is determined by its mass density $|w|^2$ modulo a global unimodular phase factor. This answers a question recently posed by Jaming for a large class of potentials, and recovers his earlier result for the free Schrödinger evolution. Our proof is simple and robust. It combines the basic local smoothing properties of the above Schrödinger evolutions, Strichartz estimates, and a stripwise version of a unique continuation theorem of Ionescu and Kenig to show that the interaction function $$ F(t,x,y)=u(t,x)v(t,y)-u(t,y)v(t,x) $$ vanishes identically on $[0,T]\times\mathbb{R}^2$ whenever $u$ and $v$ are $L^2$ solutions to a Schröinder euqation with potential $V$ with the same mass density. The method immediately extends to several important nonlinear models such as the cubic NLS equation, which is globally well-posed in $L^2(\mathbb{R})$ with $V=|u|^2$. Interestingly, we also show that the above phase recovery property fails in dimensions two and higher, even for the free equation with Schwartz initial data.

math.AP↗

arXiv2026

On the effective sharp stability of the Faber-Krahn inequality

We prove the sharp quantitative Faber--Krahn inequality with a computable dimensional constant, resolving a problem posed by Brasco and De Philippis~\cite[Open Problem~2]{brasco2017spectral}. In contrast with the known proof of the sharp estimate, our argument avoids indirect compactness arguments and selection principles. It runs through a new level-set proof of the Saint--Venant inequality, using that the torsional deficit controls both the average isoperimetric deficit and the oscillation of the gradient magnitude along the boundaries of level sets. An effective local strong isoperimetric estimate for confined sets then yields the result.

math.AP↗

arXiv2026

Eigenfunctions, free boundaries, and time-frequency localization

We develop an inverse theory for time--frequency localization operators, formulated as a free-boundary problem in which the localization domain is unknown and its boundary is recovered from prescribed spectral data. The guiding principle is that an eigenfunction can itself be regarded as geometric data determining a localization domain, with strong consequences for the associated variational problem. We obtain five main results. First, for every polynomial $f_0$ sufficiently close to the Gaussian and every $λ\in(0,1)$, we construct a real-analytic domain $U_λ$ such that $f_0$ is an eigenfunction of the corresponding localization operator with eigenvalue $λ$, providing a general inverse construction of localization domains. Second, we recover the null-set invariant Abreu--D"orfler characterization of disks as the only simply connected localization domains having a Hermite polynomial as eigenfunction. Third, we show that simple connectivity is essential: a weighted Hele--Shaw flow yields an infinite-dimensional family of non-radial, doubly connected analytic domains for which the Gaussian is an eigenfunction. Fourth, we prove the optimality of the exponent $1/2$ in the quantitative stability inequality of G'omez--Guerra--Ramos--Tilli. Finally, we prove that every local maximizer of the Gaussian Faber--Krahn problem is a disk and, using a Fock-space concentration--compactness profile decomposition, obtain a new proof of the Nicola--Tilli concentration theorem.

math.AP↗

arXiv2026

A complex-analytic proof of square-restricted stable phase retrieval in Fock space

We give a short complex-analytic proof of a square-restricted form of local stable phase retrieval at the Gaussian in one-dimensional Fock space. The main estimate is a coercivity inequality for the map $F\mapsto F^2$: \[ \|F^2-F(0)^2|_{\mathcal{F}^2(\mathbb{C})} \lesssim \inf_{c\in\mathbb{R}}\||F|^2-c\|_{L^2(d γ)}. \] The proof uses a weighted derivative norm, two integrations by parts, and a weighted Cauchy inequality. The proof has been completely verified in Lean with the aid of Large Language Models.

math.CV↗

arXiv2026

Quantitative stability for the Brascamp-Lieb inequality and moment measures

We develop a quantitative stability theory for moment measures based on a new sharp uniform stability principle for the Brascamp-Lieb variance inequality in terms of the $L^1$-distance. Our results yield structural stability estimates for solutions of the moment-measure problem that are uniform over a natural class of convex functions, thereby addressing several questions that have been open in this direction. A key novelty of our approach is that the Brascamp-Lieb stability bound is not only sharp in its stability exponent, but also uniform across a broad class of convex potentials. This uniformity is absent from previous results in the literature and, beyond its intrinsic mathematical interest, it is the mechanism that allows stability of the Brascamp-Lieb inequality to transfer to nonlinear variational problems such as the moment-measure problem. We moreover show that the $L^1-$nature of this stability estimate is sharp, in the sense that such a uniform estimate cannot hold in any $L^p$-metric, with $p>1$.

math.FA↗

arXiv2026

Stable Phase Retrieval for Spans of Independent Random Variables

We prove that, after $L^2$ normalization, stable phase retrieval holds over the $L^2$-spans of independent real-valued centered random variables if and only if all but possibly one coordinate satisfies a uniform two-sided $L^1$ bound. This provides a complete characterization of stable phase retrieval for such subspaces, building upon the pioneering work of Calderbank--Daubechies--Freeman--Freeman and confirming the conjectured characterization communicated to us by those authors. We provide two different proofs of this fact, both based on a decomposition of the $\ell^2$-coefficients of each random variable. The first is a compactness proof, which makes use of the infinite divisibility of limit laws of tail sums. The second is a quantitative proof, which substitutes the compactness step with an explicit dichotomy based on anticoncentration estimates of Sperner type. This latter proof was partially LLM generated based on the ideas in the first proof and a considerable amount of guidance by the authors. An autoformalization of our main result in Lean 4 is also provided, following the ideas in the quantitative proof.

math.FA↗

arXiv2026

$L^2$-Stability for STFT phase retrieval

We prove that the short-time Fourier transform with Gaussian window performs $L^2$-local stable phase retrieval at the constant function. The proof involved significant interplay between mathematicians and LLMs. An autoformalization in Lean 4 of an extension of our result to $L^2$-local stable phase retrieval for all Hermite windows and all elements in the finite span of the canonical basis vectors is also presented.

math.FA↗

arXiv2026

Quantitative Stability for Minkowski's problem

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1 \le p \neq n$. We prove that, for every pair of probability measures $μ,ν$ satisfying a quantitative form of the classical dispersion assumptions yielding existence of such bodies, we have a control of the form \[ \inf_{x\in \mathbb{R}^n}\mathrm{d_H}(E_μ, x + E_ν) \le C \mathrm{d_C}(μ,ν)^{\frac{1}{n-1}}, \quad α(E_μ, E_ν)^2 \le C \mathrm{d_C}(μ,ν)^{1 + \frac{1}{n-1}}, \] where $\mathrm{d_H}$ denotes the Hausdorff distance, $α$ denotes the Fraenkel asymmetry and $\mathrm{d_C}$ is the dual-convex distance of probability measures on the sphere. Our arguments are based on a variational problem whose optimizers are Minkowski bodies, for which we can obtain strong-concavity properties with the quantitative Brunn-Minkowski and isoperimetric inequalities. While the exponent in the Hausdorff distance is sharp, the exponent in the Fraenkel asymmetry is optimal in dimension $2$.

math.AP↗

arXiv2025

On sharp stable recovery from clipped and folded measurements

We investigate the stability of vector recovery from random linear measurements which have been either clipped or folded. This is motivated by applications where measurement devices detect inputs outside of their effective range. As examples of our main results, we prove sharp lower bounds on the recovery constant for both the declipping and unfolding problems whenever samples are taken according to a uniform distribution on the sphere. Moreover, we show such estimates under (almost) the best possible conditions on both the number of samples and the distribution of the data. We then prove that all of the above results have suitable (effectively) sparse counterparts. In the special case that one restricts the stability analysis to vectors which belong to the unit sphere of $\mathbb{R}^n$, we show that the problem of declipping directly extends the one-bit compressed sensing results of Oymak-Recht and Plan-Vershynin.

cs.IT↗

arXiv2025

On the regularity of Fourier interpolation formulas

By applying new functional analysis tools in the framework of Fourier interpolation formulas, such as sc-Fredholm operators and Schauder frames, we are able to improve and refine several properties of these aforementioned formulas on the real line. As two examples of our main contributions, we highlight: (i) that we may upgrade perturbed interpolation bases all the way to the Schwartz space, which shows that even the perturbed interpolation formulas are as regular as the Radchenko-Viazovska case; (ii) that a certain subset of the interpolation formulae considered by Kulikov-Nazarov-Sodin may actually be upgraded to be convergent in the Schwartz class, giving a first partial answer to a question posed by those authors. As a final contribution of this work, we also show that, if the perturbations are sufficiently small, then even analyticity properties of the basis functions are preserved. This shows, in particular, that any function that vanishes on all but finitely many of the (perturbed) nodes is automatically analytic, a feature previously only known to hold in supercritical contexts besides the Radchenko-Viazovska case.

math.CA↗

arXiv2024

Uniform stability of concentration inequalities and applications

We prove a sharp quantitative version of recent Faber-Krahn inequalities for the continuous Wavelet transforms associated to a certain family of Cauchy wavelet windows . Our results are uniform on the parameters of the family of Cauchy wavelets, and asymptotically sharp in both directions. As a corollary of our results, we are able to recover not only the original result for the short-time Fourier transform as a limiting procedure, but also a new concentration result for functions in Hardy spaces. This is a completely novel result about optimal concentration of Poisson extensions, and our proof automatically comes with a sharp stability version of that inequality. Our techniques highlight the intertwining of geometric and complex-analytic arguments involved in the context of concentration inequalities. In particular, in the process of deriving uniform results, we obtain a refinement over the proof of a previous result by the first and fourth authors together with A. Guerra and P. Tilli, further improving the current understanding of the geometry of near extremals in all contexts under consideration.

math.FA↗

arXiv2024

On Pauli pairs and Fourier uniqueness problems

We investigate the concept of Pauli pairs and a discrete counterpart to it. In particular, we make substantial progress on the question of when a discrete Pauli pair is automatically a classical Pauli pair. Effectively, if one of the functions has space and frequency Gaussian decay, and one has that $|f| = |g|$ and $|\widehat{f}| = |\widehat{g}|$ on two sets which accumulate like suitable small multiples of $\sqrt{n}$ at infinity, then $|f| \equiv |g|$ and $|\widehat{f}| = |\widehat{g}|.$ Furthermore, we show that if one drops either the assumption that one of the functions has space-frequency decay or that the discrete sets accumulate at a high rate, then the desired property no longer holds. Our techniques are inspired by and directly connected to several recent results in the realm of Fourier uniqueness problems, and our results may be seen as a nonlinear generalization of those. As a consequence of said techniques, we are able to prove a sharp discrete version of Hardy's uncertainty principle.

math.CA↗

arXiv2024

Perturbed lattice crosses and Heisenberg uniqueness pairs

This work focuses on two questions raised by H. Hedenmalm and A. Montes-Rodríguez on Heisenberg Uniqueness Pairs for perturbed lattice crosses. The first of them deals with a complete characterization of $β>0$ for which, for a fixed $θ\in \mathbb{R},$ the translated lattice cross $Λ_β^θ = ((\mathbb{Z} + \{θ\}) \times \{0\}) \cup (\{0\} \times β\mathbb{Z})$ satisfies that $(Γ,Λ_β^θ)$ is a Heisenberg Uniqueness Pair, where $Γ$ is the hyperbola in $\mathbb{R}^2$ with axes as asymptotes. We show that $(Γ,Λ_β^θ)$ is a Heisenberg Uniqueness Pair if and only if $β\le 1$, confirming a prediction made by Hedenmalm and Montes-Rodríguez. Furthermore, under modified decay conditions on the measures under consideration, we are able to prove sharp results for when a perturbed lattice cross $Λ_{\bf A,B}$ is such that $(Γ,Λ_{\bf A,B})$ is a Heisenberg Uniqueness Pair. In particular, under such decay conditions, this solves another question posed by Hedenmalm and Montes-Rodríguez. Our techniques run through the analysis of the action of the operator that maps the Fourier transform of an $L^1$ function $ψ$ to the Fourier transform of $t^{-2} ψ(1/t)$. In other words, we analyze the operator taking the restriction to the $x$-axis of a solution $u$ to the Klein-Gordon equation to its restriction to the $y$-axis. This operator turns out to be related to the action of the four-dimensional Fourier transform on radial functions, which enables us to use the framework and techniques of discrete uncertainty principles for the Fourier transform.

math.CA↗

arXiv2024

Improved stability versions of the Prékopa-Leindler inequality

We consider the problem of stability for the Prékopa-Leindler inequality. Exploiting properties of the transport map between radially decreasing functions and a suitable functional version of the trace inequality, we obtain a uniform stability exponent for the Prékopa-Leindler inequality. Our result yields an exponent not only uniform in the dimension but also in the log-concavity parameter $τ= \min(λ,1-λ)$ associated with its respective version of the Prékopa-Leindler inequality. As a further application of our methods, we prove a sharp stability result for log-concave functions in dimension 1, which also extends to a sharp stability result for log-concave radial functions in higher dimensions.

math.FA↗

arXiv2024

On Extremal Problems Associated with Random Chords on a Circle

Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius $r, \, r \in (0,1]$, where the endpoints of the chords are drawn according to a given probability distribution on $\mathbb{S}^1$. We show that, for $r=1,$ the problem is degenerated in the sense that any continuous measure is an extremiser, and that, for $r$ sufficiently close to $1,$ the desired maximal value is strictly below the one for $r=1$ by a polynomial factor in $1-r.$ Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is $1/4$ for $r \in (0,1/2).$ Connections with other variational problems and energy minimization problems are also presented.

math.MG↗

arXiv2024

A quantitative stability result for the sphere packing problem in dimensions 8 and 24

We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is $\sim \varepsilon$ close to satisfying the optimal density, then it is, in a suitable sense, $O(\varepsilon^{1/2})$ close to the $E_8$ and Leech lattices, respectively. In the periodic setting, we prove that, under the same assumptions, we may take a large 'frame' through which our packing locally looks like $E_8$ or $Λ_{24}.$ Our methods make explicit use of the magic functions constructed by M. Viazovska in dimension 8 and by H. Cohn, A. Kumar, S. Miller, the second author, and M. Viazovska in dimension 24, together with results of independent interest on the abstract stability of the lattices $E_8$ and $Λ_{24}.$

math.MG↗

arXiv2023

New Sign Uncertainty Principles

We prove new sign uncertainty principles which vastly generalize the recent developments of Bourgain, Clozel & Kahane and Cohn & Gonçalves, and apply our results to a variety of spaces and operators. In particular, we establish new sign uncertainty principles for Fourier and Dini series, the Hilbert transform, the discrete Fourier and Hankel transforms, spherical harmonics, and Jacobi polynomials, among others. We present numerical evidence highlighting the relationship between the discrete and continuous sign uncertainty principles for the Fourier and Hankel transforms, which in turn are connected with the sphere packing problem via linear programming. Finally, we explore some connections between the sign uncertainty principle on the sphere and spherical designs.

math.CA↗

arXiv2023

Stability of the Faber-Krahn inequality for the Short-time Fourier Transform

We prove a sharp quantitative version of the Faber--Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit $δ(f;Ω)$ which measures by how much the STFT of a function $f\in L^2(\mathbb R)$ fails to be optimally concentrated on an arbitrary set $Ω\subset \mathbb R^2$ of positive, finite measure. We then show that an optimal power of the deficit $δ(f;Ω)$ controls both the $L^2$-distance of $f$ to an appropriate class of Gaussians and the distance of $Ω$ to a ball, through the Fraenkel asymmetry of $Ω$. Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.

math.CA↗