Search arXiv⌕ Search

arXiv subjects

Vignon Oussa

Publications and source records attributed to Vignon Oussa.

At least 19 recordsLinked to original sources
arXiv2026

The critical mixed-arithmetic four-point HRT theorem

Although the Heil--Ramanathan--Topiwala conjecture is false in full generality, its failure makes the classification of positive geometric and arithmetic regimes more urgent. We settle the critical mixed-arithmetic regime for four time-frequency shifts. For \(z=(x,ω)\) and \(w=(y,η)\), set \(σ(z,w)=xη-yω\). Let \(u,v\in\R^2\) satisfy \(\lvertσ(u,v)\rvert=1\), write \(ν=αu+βv\), and assume that \(0,u,v,ν\) are distinct. If \(\dim_{\Q}\operatorname{span}_{\Q}\{1,α,β\}=2\) then, for every nonzero \(f\in L^2(\R)\), the vectors \(f,π(u)f,π(v)f,π(ν)f\) are linearly independent. This includes both four-point configurations in Chris Heil's Conjecture~9.2. The proof first converts a putative dependence into a scalar cocycle over an irrational rotation. On a positive-measure family of zero-free fibres, winding and continued-fraction returns force the periodic holonomy to be constant. The return multiplier is Laurent polynomial, however, so its ungauged holonomy is algebraic; the constant-holonomy law simultaneously makes it an irrational character. These conclusions are incompatible. The principal theorem and its complete formal dependency chain have been fully certified end-to-end in Lean~4, including the physical all-nonzero case, the complete three-point HRT theorem, the coefficient-reduction step, and the final four-point conclusion. The accompanying \href{https://www.dropbox.com/scl/fi/jd9xeqqera147wdc82r8k/3f260460-07d5-45f5-8d1c-ee1d2879c0af-aristotle-32-.tar.gz?rlkey=a24njadfds5ty9ez473lk03lv\&e=1\&dl=0} {Lean~4 certification archive} contains reproducible source, pinned build instructions, and transitive axiom audits.

math.CA↗
arXiv2026

A minimal HRT counterexample at the sharp Gelfand--Shilov threshold

In a remarkable breakthrough, Faulhuber, Petersen, van Velthoven, and Voigtlaender disproved the Heil--Ramanathan--Topiwala conjecture by constructing twelve linearly dependent time--frequency shifts of a Schwartz function. The following day, the author compressed their vector-Zak and cohomological architecture to an explicit intrinsically subcritical four-point configuration, thereby attaining the minimum possible cardinality. Soon thereafter, Jasper and Mixon gave a transparent analytic refinement of the breakthrough mechanism and proved, for every Diophantine pair, that HRT counterexamples may be realized with windows in the Roumieu Gelfand--Shilov class $\mathcal S_1^1(\mathbb R)$; their theorem establishes the endpoint regularity phenomenon in broad arithmetic generality while leaving support cardinality unrestricted. The purpose of the present paper is to show that these two sharp advances can be attained simultaneously. We prove that the same explicit four points, coefficients, and matrix cocycle admit a nonzero window in $\mathcal S_1^1(\mathbb R)$. Moreover, if $0\ne f\in\mathcal S_s^σ(\mathbb R)$ and $s<1$ or $σ<1$, then every finite system of distinct time--frequency shifts of $f$ is linearly independent. Thus four is the least possible cardinality and $(1,1)$ is the sharp coordinatewise threshold within the Roumieu Gelfand--Shilov scale.

math.CA↗
arXiv2026

Near-Parseval orbit frames for irreducible unitary representations: from mixing and expansion

We establish an abstract storage theorem for projectively $C_0$ unitary representations admitting a pair of one-parameter subgroups with conjugation escape. It produces, for every $0<\varepsilon<1$, a single orbit whose sampling set is relatively separated modulo the projective kernel and whose frame bounds are $(1-\varepsilon)^2$ and $(1+\varepsilon)^2$. Two applications are obtained from the same mechanism. First, every nonabelian exponential Lie group admits a near-Parseval frame of left translates for its left regular representation. Consequently, a positive-dimensional exponential Lie group is an FT group if and only if it is nonabelian; in particular, the three-dimensional Heisenberg group admits such a frame. Second, every infinite-dimensional irreducible unitary representation of an exponential Lie group admits a near-Parseval discrete orbit frame. When the effective projective quotient is abelian, the orbit may be chosen to be an orthonormal basis arising from a transported Weyl lattice.

math.FA↗
arXiv2026

An intrinsically subcritical four-point counterexample

Building on the vector-Zak and cohomological framework developed by Faulhuber, Petersen, van Velthoven, and Voigtlaender in their twelve-point counterexample, we give a computer-assisted four-point counterexample with a nonzero complex-valued Schwartz window. Every symplectic triangle determinant of the explicit configuration has absolute value below one, placing it in the intrinsically subcritical regime.

math.CA↗
arXiv2026

Lean-certified four-point HRT results for three lattice points and one off-lattice point

We record a Lean-certified theorem package for the four-point Heil--Ramanathan--Topiwala configuration \[ Λ=\{0,a,b,ν\}\subset \R^2, \qquad \Lzero=\Z a+\Z b, \qquad ν=r a+s b, \] with $a$ and $b$ linearly independent. The principal certified theorem states that if $|\symp(a,b)|>1$ and $1,r,s$ are linearly independent over $\Q$, then for every nonzero $f\in L^2(\R)$ the four vectors \[ f,\qquad π(a)f,\qquad π(b)f,\qquad π(ν)f \] are linearly independent. A second certified theorem treats the rational-coordinate case $r,s\in \Q$, where the configuration lies in a finer full-rank lattice and linear independence follows from Linnell's theorem. The paper is written in standard mathematical prose. An appendix records the precise Lean certification ledger and the explicit analytic inputs used by the formal development and a download link is provided.

math.FA↗
arXiv2026

Trichotomy for the HRT Conjecture for mixed integer configuration

We consider the HRT conjecture in the mixed-integer setting, where a finite configuration in $\R^d\times\R^d$ consists of $N-1$ points in $\Z^d\times\Z^d$ and one point $(α,β)$ outside the lattice. Assuming a linear dependence among the corresponding time-frequency shifts of a nonzero Schwartz function, we apply the Zak transform to obtain a cocycle over translation by $γ=(-α,β)$ on $\T^{2d}$ and study the orbit closure \[ H=\overline{\{nγ\bmod \Z^{2d}:n\in\Z\}}. \] We show that this reduction yields a trichotomy. The dense-orbit case is impossible because a Zak zero propagates to a dense zero set, forcing the Zak transform to vanish identically. The finite-orbit case reduces to a rational configuration, and hence to the lattice case covered by Linnell's theorem. Thus any mixed-integer counterexample for a Schwartz window must occur in the infinite proper case. For that remaining case, we prove that the nonvanishing set of the Zak transform is $H$-saturated, that the averaged logarithmic growth of the modulus cocycle along $H$ exists and vanishes identically, and that the restriction to each nonvanishing $H_0$-coset satisfies a smooth cohomological equation. This yields small-divisor compatibility conditions for the induced translation on $H_0$. We further obtain an arithmetic rigidity condition. These results isolate a collection of necessary dynamical, cohomological, and arithmetic constraints that any mixed-integer counterexample must satisfy.

math.DS↗
arXiv2025

Translation--Modulation Identities, Ergodic Log-Products and a Conditional Obstruction for Schwartz Functions

We study very smooth functions on the real line, namely Schwartz functions, that satisfy a finite identity relating their translates and a single modulation. Concretely, we assume there is a nontrivial linear combination of translates of the function that equals a fixed frequency shift of the same function. Passing to the Fourier transform turns this into a multiplicative transfer relation: the value of the Fourier transform at one point is obtained by multiplying its value at another point by a trigonometric polynomial. Iterating this relation expresses the Fourier transform along an arithmetic progression as a product of such trigonometric factors times a fixed initial value. We then recast this product in an ergodic theoretic framework by viewing it as a Birkhoff sum for a continuous observable on a compact abelian group generated by a diagonal unitary matrix. The key quantity controlling the growth or decay of the Fourier transform along the progression is a space average, namely the integral over the compact group of the logarithm of the absolute value of the associated trigonometric polynomial. The main rigorously proved statement is the following. If, for some frequency where the Fourier transform does not vanish along the entire progression, this space average is nonzero, or if the average is zero but a certain recurrence set, in the sense of Atkinson's theorem, intersects a distinguished one-parameter subgroup, then the assumed translation--modulation identity forces exponential growth of the Fourier transform along some sequence of points. This contradicts the rapid decay required of a Schwartz function.

math.FA↗
arXiv2024

The HRT conjecture for two classes of special configurations

The HRT (Heil-Ramanathan-Topiwala) conjecture stipulates that the set of any finitely many time-frequency shifts of a non-zero square Lebesgue integrable function is linearly independent. The present work settles two special cases of this conjecture, namely, the cases where the set of time-frequency shifts has cardinality $N+1$ such that either $N$ of the points lie on some integer lattice and the last point is arbitrary, or $N$ of the points are on a line, while the last point does not belong this line. In both cases, we prove that the HRT conjecture holds appealing mainly to various forms of the ergodic theorem. We note that, in recent years, the latter case has been the subject of many investigations -- notably, the subcase where $N=3$ -- and our work completely resolves it.

math.FA↗
arXiv2023

Zero set of Zak transform and the HRT Conjecture

The HRT (Heil-Ramanathan-Topiwala) posits the linear independence of any set of nonzero square-integrable vectors obtained from a single nonzero vector $f$ by applying a finite set of time-frequency shift operators. In this short note, we present findings centered on the zero set of the Zak transform of $f$, and a distinct arrangement involving a finite set of $N$ points on an integer lattice in the time-frequency plane, excluding a specific point.

math.FA↗
arXiv2023

Phase retrieval for affine groups over prime fields

We study phase retrieval for group frames arising from permutation representations, focusing on the action of the affine group of a finite field. We investigate various versions of the phase retrieval problem, including conjugate phase retrieval, sign retrieval, and matrix recovery. Our main result establishes that the canonical irreducible representation of the affine group $\mathbb{Z}_p \rtimes \mathbb{Z}_p^\ast$ (with $p$ prime), acting on the vectors in $\mathbb{C}^{p}$ with zero-sum, has the strongest retrieval property, allowing to reconstruct matrices from scalar products with a group orbit consisting of rank-one projections. We explicitly characterize the generating vectors that ensure this property, provide a linear matrix recovery algorithm and explicit examples of vectors that allow matrix recovery. We also comment on more general permutation representations.

math.RT↗
arXiv2023

Phase retrieval for nilpotent groups

We study the phase retrieval property for orbits of general irreducible representations of nilpotent groups, for the classes of simply connected connected Lie groups, and for finite groups. We prove by induction that in the Lie group case, all irreducible representations do phase retrieval. For the finite group case, we mostly focus on $p$-groups. Here our main result states that every irreducible representation of an arbitrary $p$-group with exponent $p$ and size $\le p^{2+p/2}$ does phase retrieval. Despite the fundamental differences between the two settings, our inductive proof methods are remarkably similar.

math.RT↗
arXiv2020

On exponential bases and frames with non-linear phase functions and some applications

In this paper, we study the spectrality and frame-spectrality of exponential systems of the type $E(Λ,φ) = \{e^{2πi λ\cdotφ(x)}: λ\inΛ\}$ where the phase function $φ$ is a Borel measurable which is not necessarily linear. A complete characterization of pairs $(Λ,φ)$ for which $E(Λ,φ)$ is an orthogonal basis or a frame for $L^{2}(μ)$ is obtained. In particular, we show that the middle-third Cantor measures and the unit disc, each admits an orthogonal basis with a certain non-linear phase. Under a natural regularity condition on the phase functions, when $μ$ is the Lebesgue measure on $[0,1]$ and $Λ= {\mathbb{Z}},$ we show that only the standard phase functions $φ(x) = \pm x$ are the only possible functions that give rise to orthonormal bases. Surprisingly, however we prove that there exist a greater degree of flexibility, even for continuously differentiable phase functions in higher dimensions. For instance, we were able to describe a large class of functions $φ$ defined on ${\mathbb{R}}^{d}$ such that the system $E(Λ,φ)$ is an orthonormal basis for $L^{2}[0,1]^{d}$ when $d\geq2.$ Moreover, we discuss how our results apply to the discretization problem of unitary representations of locally compact groups for the construction of orthonormal bases. Finally, we conclude the paper by stating several open problems.

math.FA↗
arXiv2020

Full Spark Frames in the Orbit of a Representation

We present a new infinite family of full spark frames in finite dimensions arising from a unitary group representation, where the underlying group is the semi-direct product of a cyclic group by a group of automorphisms. The only previously known algebraically constructed infinite families were the harmonic, Gabor and Dihedral frames. Our construction hinges on a theorem that requires no group structure. Additionally, we illustrate our results by providing explicit constructions of full spark frames.

math.RT↗
arXiv2018

Compactly supported bounded frames on Lie groups

Let $G=NH$ be a Lie group where $N,H$ are closed connected subgroups of $G,$ and $N$ is an exponential solvable Lie group which is normal in $G.$ Suppose furthermore that $N$ admits a unitary character $χ_λ$ corresponding to a linear functional $λ$ of its Lie algebra. We assume that the map $h\mapsto Ad\left( h^{-1}\right) ^{\ast}λ$ defines an immersion at the identity of $H$. Fixing a Haar measure on $H,$ we consider the unitary representation $π$ of $G$ obtained by inducing $χ_λ.$ This representation which is realized as acting in $L^{2}\left( H,dμ_{H}\right) $ is generally not irreducible, and we do not assume that it satisfies any integrability condition. One of our main results establishes the existence of a countable set $Γ\subset G$ and a function $\mathbf{f}\in L^{2}\left( H,dμ_{H}\right) $ which is compactly supported and bounded such that $\left\{ π\left( γ\right) \mathbf{f}:γ\inΓ\right\} $ is a frame. Additionally, we prove that $\mathbf{f}$ can be constructed to be continuous. In fact, $\mathbf{f}$ can be taken to be as smooth as desired. Our findings extend the work started in \cite{oussa2018frames} to the more general case where $H$ is any connected Lie group. We also solve a problem left open in \cite{oussa2018frames}. Precisely, we prove that in the case where $H$ is an exponential solvable group, there exist a continuous (or smooth) function $\mathbf{f}$ and a countable set $Γ$ such that $\left\{ π\left( γ\right) \mathbf{f}:γ\inΓ\right\} $ is a Parseval frame. Since the concept of well-localized frames is central to time-frequency analysis, wavelet, shearlet and generalized shearlet theories, our results are relevant to these topics and our approach leads to new constructions which bear potential for applications.

math.RT↗
arXiv2018

Groups with frames of translates

Let $G$ be a locally compact group with left regular representation $λ_{G}.$ We say that $G$ admits a frame of translates if there exist a countable set $Γ\subset G$ and $φ\in L^{2}(G)$ such that $(λ_{G}(x) φ)_{x \inΓ}$ is a frame for $L^{2}(G).$ The present work aims to characterize locally compact groups having frames of translates, and to this end, we derive necessary and/or sufficient conditions for the existence of such frames. Additionally, we exhibit surprisingly large classes of Lie groups admitting frames of translates.

math.RT↗
arXiv2017

Dihedral Group Frames with the Haar Property

We consider a unitary representation of the Dihedral group $D_{2n}% =\mathbb{Z}_{n}\rtimes\mathbb{Z}_{2}$ obtained by inducing the trivial character from the co-normal subgroup $\left\{0\right\}\rtimes\mathbb{Z}_{2}.$ This representation is naturally realized as acting on the vector space $\mathbb{C}^{n}.$ We prove that the orbit of almost every vector in $\mathbb{C}^{n}$ with respect to the Lebesgue measure has the Haar property (every subset of cardinality $n$ of the orbit is a basis for $\mathbb{C}^{n}$) if $n$ is an odd integer. Moreover, we provide explicit sufficient conditions for vectors in $\mathbb{C}^{n}$ whose orbits have the Haar property. Finally, we derive that the orbit of almost every vector in $\mathbb{C}^{n}$ under the action of the representation has the Haar property if and only if $n$ is odd. This completely settles a problem which was only partially answered in \cite{Oussa}.

math.FA↗
arXiv2017

Frames arising from irreducible solvable actions Part I

Let $G$ be a simply connected, connected completely solvable Lie group with Lie algebra $\mathfrak{g}=\mathfrak{p}+\mathfrak{m}.$ Next, let $π$ be an infinite-dimensional unitary irreducible representation of $G$ obtained by inducing a character from a closed normal subgroup $P=\exp\mathfrak{p}$ of $G.$ Additionally, we assume that $G=P\rtimes M,$ $M=\exp\mathfrak{m}$ is a closed subgroup of $G,$ $dμ_{M}$ is a fixed Haar measure on the solvable Lie group $M$ and there exists a linear functional $λ\in\mathfrak{p}^{\ast}$ such that the representation $π=π_λ=\mathrm{ind}_{P}^{G}\left( χ_λ\right) $ is realized as acting in $L^{2}\left( M,dμ_{M}\right) .$ Making no assumption on the integrability of $π_λ$, we describe explicitly a discrete subgroup $Γ\subset G$ and a vector $\mathbf{f}\in L^{2}\left( M,dμ_{M}\right) $ such that $π_{λ}\left( Γ\right) \mathbf{f}$ is a tight frame for $L^{2}\left( M,dμ_{M}\right) .$ We also construct compactly supported smooth functions $\mathbf{s}$ and discrete subsets $Γ\subset G$ such that $π_{λ}\left( Γ\right) \mathbf{s}$ is a frame for $L^{2}\left( M,dμ_{M}\right) .$

math.FA↗