Fourier Nonuniqueness Pairs

• Fourier nonuniqueness pairs are pairs of sets where a nonzero function or measure vanishes on one set while its Fourier transform vanishes on the other, challenging the injectivity of the Fourier mapping.
• They are constructed via methods ranging from the Poisson Summation Formula to modular forms, yielding both periodic lattice-like and aperiodic quasicrystal structures.
• The concept has significant implications for phase retrieval, interpolation formulas, and density threshold analysis, impacting both theoretical exploration and practical signal processing.

A Fourier nonuniqueness pair consists of two sets (or sequences) such that there exists a nonzero function or measure vanishing on one set whose Fourier transform vanishes on the other. In this situation, the mapping between a function and its sampled values (or support) and the corresponding data for its Fourier transform becomes non-injective. Fourier nonuniqueness pairs are a fundamental concept in harmonic analysis, with deep implications for phase retrieval, interpolation formulas, and rigidity phenomena in real and Fourier space.

1. Formal Definitions and Examples

A pair of closed sets (S, T) ⊆ ℝ × ℝ is called a Fourier nonuniqueness pair if there exists a nonzero (tempered) measure μ with $\operatorname{supp} μ⊆S$ and $\operatorname{supp} \widehat{μ}⊆T$ (Kolountzakis, 2015). Analogously, for function classes (e.g., Schwartz functions), a pair $(\Lambda, W)$ of discrete sets is nonuniqueness if there exists $f\not\equiv0$ with $f(\lambda) = 0$ for all $\lambda\in\Lambda$ and $\widehat{f}(w) = 0$ for all $w\in W$ (Kulikov et al., 2023). In the context of phase retrieval, nonuniqueness pairs (also called ambiguity pairs) are discrete signals $x, y$ such that $|X(e^{j\omega})| = |Y(e^{j\omega})|$ for all $\omega$, though $x\not\propto y$ (Bendory et al., 2017).

A classical example is given by two functions with disjoint Fourier supports; for any $c\in\mathbb{S}^1$, $F = f_1 + c f_2$ and $G = f_1 + f_2$ satisfy $|\widehat{F}|=|\widehat{G}|$, yet $F\not\propto G$ (Andreys et al., 2015).

2. Structural Constructions and Quasicrystals

Traditional constructions of nonuniqueness pairs on ℝ were derived from the Poisson Summation Formula (PSF), yielding measures whose support and the support of their Fourier transforms are contained in finite unions of arithmetic progressions or lattices. Translating, modulating, or dilating the standard Dirac comb produces such measures.

However, (Kolountzakis, 2015) established the existence of discrete measures on the real line whose Fourier transform is also discretely supported, but where neither support is contained in a finite union of arithmetic progressions. The construction uses periodic measures with increasingly large gaps for both a measure and its Fourier transform, then superposes these blocks using shifts that are linearly independent over ℚ, yielding a set spanning an infinite-dimensional ℚ-vector space. The resulting supports can be thought of as "quasicrystals" without lattice structure. This disproves the idea that all discrete Fourier pairs originate from iterative applications of the PSF, revealing the existence of aperiodic, arithmetic-structureless nonuniqueness pairs.

3. Quantitative Criteria and Critical Density

Kulikov–Nazarov–Sodin (Kulikov et al., 2023) gave sharp, near-matching density and spacing criteria distinguishing uniqueness from nonuniqueness on the real line. For two discrete sets Λ, M ordered on ℝ, define

$$D_\Lambda = \liminf_{|j|\to\infty} |\lambda_j|^{p-1} (\lambda_{j+1} - \lambda_j),$$

for $1<p,q<\infty$, $1/p+1/q=1$. If $D_\Lambda > 1$ and similarly for $D_M$, then for each such subcritical pair $(\Lambda, M)$ there exists $f \in S(p,q)$ vanishing on $\Lambda$ with $\widehat{f}$ vanishing on $M$—a nonuniqueness pair. If instead the limsup is <1, uniqueness holds.

This criterion refines the intuition that sampling beyond a critical density secures uniqueness: above threshold, there is nonuniqueness; below, uniqueness and even interpolating frame formulas. The construction uses entire functions with prescribed zeros (Hadamard–Brelot type), and the Gelfand–Shilov space provides the ambient class for these results.

4. Nonuniqueness in Phase Retrieval and Modulus Constraints

In phase retrieval, a nonuniqueness (Pauli) pair is a pair $(f,g)$ such that $|f(x)|=|g(x)|$ and $|\widehat{f}(\xi)|=|\widehat{g}(\xi)|$ almost everywhere. Discretizing, one considers $(\Lambda, \Gamma)$ and asks under what conditions modulus data at accumulating points uniquely determines the function up to global phase. Ramos–Sousa (Ramos et al., 2024) established that if one partner satisfies Gaussian decay and $\Lambda, \Gamma$ densify at rate $\sim 1/|\lambda|$, then matching modulus data suffices for uniqueness; but if either the decay or this accumulation is lost, infinite-dimensional nonuniqueness arises. This describes the transition between uniqueness and severe nonuniqueness for modulus-type constraints.

In the discrete Fourier case, the ambiguity polynomial structure shows that, up to trivial ambiguities (global phase, shift, conjugate reflection), the quotient of nonuniqueness is classified by binary choices at reciprocal-conjugate pairs of zeros of the autocorrelation polynomial (Bendory et al., 2017).

5. Nonuniqueness Pairs on Algebraic and Geometric Structures

In higher dimensions, (Radchenko et al., 2021) constructed infinite-dimensional vanishing spaces for Schwartz eigenfunctions on ℝⁿ vanishing on the componentwise square roots of certain algebraic lattices (from totally real number fields) or related ellipsoidal sets, revealing geometric sources of nonuniqueness. The counting of such sets demonstrates that they can fail minimality by a large excess compared to known uniqueness sets.

For curves such as the hyperbola, nonuniqueness arises in the form of critical density phenomena. For a hyperbola branch, at critical sampling rates (determined by cross-lattice parameters), one finds a one-dimensional nullspace of measures whose Fourier transform vanishes on the given lattice-cross, but any further removal of nodes destroys this flexibility (Hedenmalm, 2011). This nuance is linked to ergodic and transfer-operator arguments (via Perron–Frobenius operators) and has connections to the Nielsen spiral in the Klein–Gordon setting.

6. Interpolation Formulas, Excess, and Limiting Behavior

The existence and structure of Fourier nonuniqueness pairs are closely connected to the theory of discrete Fourier interpolation. For even Schwartz functions, the Radchenko–Viazovska interpolation formula yields an explicit uniqueness pair at critical density (e.g., $\{\sqrt{n}\}_{n=0}^\infty$), but (Berghaus et al., 21 Dec 2025) demonstrates that by removing a small number of nodes and adding many "extraneous zeros" (e.g., zeros of an explicit modular-form-based basis function), nonuniqueness pairs of arbitrarily large excess can be constructed. The excess, defined as $e - r$ where $e$ is the number of additional nodes and $r$ nodes are removed, can grow like $\sqrt{n}/\log^2 n$. The analysis of the distribution and asymptotics of these new zeros elucidates the limits of uniqueness and interpolation framed by modular forms.

Numerically, as $n\to\infty$, the basis function $f_n$ possesses an increasing number of zeros ("extraneous nodes") beyond those determined by the minimal uniqueness set, leading to families of nonuniqueness at the critical density threshold. The failure of these basis functions to form a Riesz basis in certain Hilbert spaces further underscores the delicate nature of the interpolation–nonuniqueness dichotomy (Berghaus et al., 21 Dec 2025).

7. Applications and Open Directions

Fourier nonuniqueness pairs arise throughout signal processing (especially in phase retrieval and spectrogram recovery (Grohs et al., 2022)), additive and combinatorial harmonic analysis, and the study of quasicrystals. Their construction leverages a diverse array of analytic, algebraic, and combinatorial techniques: modular forms, ergodic theory, automorphic representation, entire function growth, and operator theory.

Open questions include full classification of pairs arising from PSF versus genuinely aperiodic constructions, characterization of extra hypotheses (density, decay) ensuring uniqueness, extension of the theory to higher dimensions and non-Euclidean settings, and links to other transforms (Hankel, Radon) (Kolountzakis, 2015, Kulikov et al., 2023, Radchenko et al., 2021). The interplay of algebraic and geometric structure—especially in relation to totally real fields, Hecke groups, and modular forms—remains a central theme, as does the study of the sharpness of uniqueness thresholds and the construction of crystalline or quasicrystalline measures.

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Table: Principal Settings for Fourier Nonuniqueness Pairs

Setting	Type of Nonuniqueness Pair	Reference
Discrete supports in ℝ	Quasicrystal without AP structure	(Kolountzakis, 2015)
Critical/interpolating node sets	Large excess via extraneous zeros	(Berghaus et al., 21 Dec 2025)
Phase retrieval in L²(ℝ)	Ambiguity pairs, modulus-only	(Bendory et al., 2017, Ramos et al., 2024)
Lattices from number fields (ℝⁿ)	Infinite-dimensional vanishing	(Radchenko et al., 2021)
Hyperbola in ℝ²	Critical defect (density)	(Hedenmalm, 2011, Giri, 2020)
STFT/spectrogram discrete sampling	Nonuniqueness for any lattice	(Grohs et al., 2022)

Every setting illuminates both classical and modern phenomena: where sampling theorems, phase retrieval, and the geometry of function spaces meet the fundamental rigidity—or failure thereof—of the Fourier transform.