Fourier Uniqueness Pairs

• Fourier uniqueness pairs are defined as configurations in which a function or measure is uniquely determined when its Fourier transform vanishes on a paired set.
• They extend to various settings including planar curves, quadratic hypersurfaces, and discrete sampling, with results that highlight critical density and symmetry thresholds.
• The theory leverages affine invariance, dynamical systems, and interpolation methods to unify classical harmonic analysis and PDE uniqueness results.

Fourier uniqueness pairs, often called Heisenberg uniqueness pairs, are configurations in which simultaneous information in physical space and Fourier space determines a function or measure uniquely. In one standard measure-theoretic formulation, measurable sets $S,\Lambda\subseteq\mathbb{R}^d$ form a Heisenberg uniqueness pair if every finite Borel measure $\mu$ supported on $S$ whose Fourier transform vanishes on $\Lambda$ is necessarily zero; in function-space formulations, a pair of discrete or continuous sets plays the same role for classes such as Schwartz, Sobolev, Hardy, or band-limited spaces. The subject now encompasses planar curves, quadratic hypersurfaces, critical-density discrete sampling, phaseless Pauli-type analogues, and group or multiplier extensions (Gröchenig et al., 2016).

1. Core definitions and formal frameworks

In the continuous Euclidean setting, the basic definition is: for measurable $S,\Lambda\subseteq \mathbb{R}^d$, let $M(S)$ denote the finite signed or complex Borel measures supported on $S$. Then $(S,\Lambda)$ is a Heisenberg uniqueness pair if

$$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$

for every $\mu\in M(S)$. Equivalently, if two finite measures supported on $\mu$0 have Fourier transforms agreeing on $\mu$1, then the measures are equal (Gröchenig et al., 2016).

A parallel function-space formulation is standard in the discrete theory. For a function class $\mu$2 on $\mu$3, a pair $\mu$4 is a uniqueness pair if

$\mu$5

for all $\mu$6; otherwise it is a non-uniqueness pair. This language is used for Schwartz spaces, Fourier-symmetric Sobolev spaces, Gelfand–Shilov spaces, and related Hilbert spaces (Kulikov et al., 2023).

Several papers emphasize that the numerical constants in density or criticality thresholds depend on the Fourier normalization. This explains why the same geometric phenomenon appears with thresholds such as $\mu$7, $\mu$8, $\mu$9, $S$0, or $S$1 in different articles: the underlying uniqueness question is invariant, but the constants rescale with the chosen convention (Lysen, 21 May 2026).

A recurrent structural feature is invariance under affine changes of variables. For Euclidean HUPs, translation and invertible linear maps preserve the property, and this allows one to normalize many geometries before analysis (Gröchenig et al., 2016).

2. Planar archetypes: hyperbolas, circles, parabolas, and dynamical systems

The foundational planar example is the hyperbola

$S$2

paired with the lattice cross

$S$3

Hedenmalm–Montes-Rodríguez proved that $S$4 is a HUP if and only if $S$5. For the one-branch hyperbola, the threshold changes: $S$6 is HUP if and only if $S$7, while at $S$8 the defect is one-dimensional [(Huang et al., 2023); (Hedenmalm, 2011)].

The Klein–Gordon connection is intrinsic. If $S$9 with $\Lambda$0 supported on $\Lambda$1, then $\Lambda$2 solves

$\Lambda$3

The lattice-cross thus becomes a uniqueness/observability set for a Fourier-supported Klein–Gordon solution. At the critical one-branch threshold, the annihilator is explicitly described, and its Fourier transform does not vanish at any additional point on the axes; this is tied to the Nielsen spiral through cosine and sine integrals (Hedenmalm, 2011).

A major conceptual advance is the dynamical-system reformulation of vanishing on lines. When $\Lambda$4 is a union of lines through the origin, vanishing of $\Lambda$5 on a line can be rewritten as a weighted antisymmetry relation across the fibers of the corresponding projection. For two lines, the resulting involutions $\Lambda$6 and $\Lambda$7 generate a map $\Lambda$8 on the parameter set of the curve, and uniqueness can follow from wandering intervals, attractive intervals, or conjugacy to an irrational rotation. This dynamical viewpoint recovers the classical circle criterion: for the unit circle and two lines through the origin, HUP holds exactly when the angle difference is not in $\Lambda$9; it also unifies proofs for circles, parabolas, and hyperbolas and produces new examples for cusps, polygons, and smooth convex curves (Jaming et al., 2013).

The endpoint case of the hyperbola family, namely

$S,\Lambda\subseteq \mathbb{R}^d$0

is qualitatively different. Here $S,\Lambda\subseteq \mathbb{R}^d$1 is the union of the coordinate axes, and any $S,\Lambda\subseteq \mathbb{R}^d$2 decomposes as

$S,\Lambda\subseteq \mathbb{R}^d$3

so that

$S,\Lambda\subseteq \mathbb{R}^d$4

For this wave-equation endpoint, density of the coordinate projections of $S,\Lambda\subseteq \mathbb{R}^d$5 is necessary, but in general not sufficient. When $S,\Lambda\subseteq \mathbb{R}^d$6 lies in the union of two graphs $S,\Lambda\subseteq \mathbb{R}^d$7 of continuous bijections $S,\Lambda\subseteq \mathbb{R}^d$8, the decisive object is $S,\Lambda\subseteq \mathbb{R}^d$9: under a mild nondegeneracy hypothesis, $M(S)$0 is HUP if and only if $M(S)$1 is dense and the open set $M(S)$2 is wandering for $M(S)$3 (Huang et al., 2023).

3. Quadratic hypersurfaces, cones, and higher-dimensional geometry

A higher-dimensional turning point is the quadratic-hypersurface theorem of Gröchenig and Jaming. Let

$M(S)$4

where $M(S)$5 is a quadratic form. Then there exists an exceptional set $M(S)$6 of measure zero such that, if $M(S)$7 are distinct, satisfy $M(S)$8, $M(S)$9, and $S$0, the pair

$S$1

is a Heisenberg uniqueness pair. The result applies in arbitrary dimension and covers, up to affine equivalence, spheres, ellipsoids, paraboloids, cones, hyperboloids, and degenerate quadratic sets (Gröchenig et al., 2016).

The proof rests on three geometric mechanisms. First, vanishing on a single hyperplane with non-isotropic normal $S$2 yields an odd symmetry under the affine involution

$S$3

Second, vanishing on two hyperplanes forces invariance under the composition of two such reflections. Third, disintegration along two-dimensional affine sections reduces the problem to conic sections in the plane, where a complete HUP classification was already available. The measure-zero exceptional set arises from the countable set of bad angles in this planar reduction (Gröchenig et al., 2016).

A sharp corollary is a Cramér–Wold theorem specialized to quadratic supports: a finite measure supported on $S$4 is uniquely determined by its projections onto two generic hyperplanes, whereas an arbitrary finite measure on $S$5 requires a dense family of projections. The same reflection mechanism also yields a unique-continuation statement for second-order constant-coefficient PDEs: if $S$6 and $S$7 vanishes on two generic hyperplanes, then $S$8 (Gröchenig et al., 2016).

The theorem is not a generic statement about arbitrary supports. For a single hyperplane $S$9, $(S,\Lambda)$0 is a HUP if and only if the orthogonal projection $(S,\Lambda)$1 is one-to-one on $(S,\Lambda)$2. In particular, if $(S,\Lambda)$3, then $(S,\Lambda)$4 is generally not a HUP; if $(S,\Lambda)$5, uniqueness depends on the isotropic set

$(S,\Lambda)$6

On spheres, the parallel-hyperplane case is especially sharp: for general finite measures on $(S,\Lambda)$7, $(S,\Lambda)$8 is a HUP if and only if $(S,\Lambda)$9, while for absolutely continuous measures it is an $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$0-HUP for any $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$1. If finitely many hyperplanes through the origin are used, the exact criterion is Coxeter-theoretic: $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$2 is a HUP if and only if the Coxeter group generated by the orthogonal reflections in the $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$3 is infinite (Gröchenig et al., 2016).

A complementary geometric theorem concerns cones. For the Euclidean Fourier transform on $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$4, $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$5 is a HUP if and only if the cone $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$6 is not contained in the zero set of any nonzero homogeneous harmonic polynomial. This gives an exact characterization of sphere–cone HUPs and connects them to sets of injectivity for the spherical mean operator. In $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$7, the paraboloid and certain circles on $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$8 are HUPs for a specific symmetric class of finite Borel measures on $$\widehat{\mu}(\xi)=0\quad\text{for all }\xi\in\Lambda \quad\Longrightarrow\quad \mu\equiv 0$$9 (Srivastava, 2015).

4. Discrete criticality, interpolation, and non-uniqueness

The discrete theory replaces geometric supports by growth and spacing conditions on sequences. In one influential framework, for ordered sequences $\mu\in M(S)$0 and $\mu\in M(S)$1 with $\mu\in M(S)$2 and $\mu\in M(S)$3, a pair is called supercritical when

$\mu\in M(S)$4

and subcritical when the corresponding liminf quantities are $\mu\in M(S)$5. Any supercritical pair is a uniqueness pair for the Schwartz space, indeed for the Fourier-symmetric Sobolev space $\mu\in M(S)$6, while any subcritical pair is a non-uniqueness pair already in a Gelfand–Shilov space $\mu\in M(S)$7 (Kulikov et al., 2023).

In the same setting, separated supercritical pairs satisfy a two-sided frame inequality in $\mu\in M(S)$8 and admit a Fourier interpolation formula

$\mu\in M(S)$9

with convergence in $\mu$00. This interpolation, in turn, produces locally finite measures with discrete support and discrete Fourier transform, giving an explicit source of crystalline measures (Kulikov et al., 2023).

The model family

$\mu$01

exhibits the threshold sharply: $\mu$02 is supercritical, $\mu$03 is subcritical, and the case $\mu$04 is the critical square-root scale associated with Radchenko–Viazovska interpolation (Kulikov et al., 2023).

The critical regime itself is subtler. A recent sufficient condition uses a subexponentially admissible weight $\mu$05 and shifted gap controls

$\mu$06

and yields Fourier uniqueness pairs in the critical case. This produces strongly asymmetric examples such as

$\mu$07

which approach the Bondarenko–Radchenko–Seip profile and move toward the asymmetry of Shannon–Whittaker sampling while remaining purely discrete on both sides (Lysen, 22 Sep 2025).

In higher dimensions, density can be expressed through variable-radius coverings. For closed discrete $\mu$08, the notions of $\mu$09-dense and $\mu$10-separated control whether the map

$\mu$11

is injective or surjective on $\mu$12. If $\mu$13 and both sets are sufficiently dense, injectivity holds for small $\mu$14; if $\mu$15 and both are sufficiently separated, the kernel is infinite-dimensional and, for large $\mu$16, arbitrary rapidly decreasing data can be prescribed on $\mu$17 and $\mu$18. At criticality $\mu$19, small-$\mu$20 injective examples arise from pseudohomogeneous deformations of lattices, while large-$\mu$21 separated sets yield infinite-dimensional kernels. A concrete example is

$\mu$22

which is a Fourier uniqueness pair for sufficiently small $\mu$23 and a non-uniqueness pair for sufficiently large $\mu$24 (Adve, 2023).

Arithmetic specializations sharpen this picture. For $\mu$25, if $\mu$26 vanishes on $\mu$27 and $\mu$28 vanishes on $\mu$29, uniqueness is proved on an explicit region

$\mu$30

with the diagonal corollary $\mu$31 (Ramos et al., 2019). A plausible implication, explicitly conjectured there, is that the full subcritical condition $\mu$32 should be sufficient.

Critical symmetry does not preclude non-uniqueness. The basis functions underlying the Radchenko–Viazovska interpolation formula on $\mu$33 have many “extraneous zeros,” and these zeros can be used to construct Fourier non-uniqueness pairs whose apparent excess over the critical pair may be made arbitrarily large. The same analysis shows that the normalized RV basis functions do not form a Riesz basis in the Kulikov–Nazarov–Sodin Hilbert space (Berghaus et al., 21 Dec 2025).

A finite cyclic analogue also exists. For the discrete Fourier transform on $\mu$34, the restricted space

$\mu$35

is governed by a commuting periodically tridiagonal matrix $\mu$36. The simple eigenspaces of $\mu$37 span $\mu$38, and the dimension formula is

$\mu$39

For $\mu$40, there is an explicit interpolation formula reconstructing $\mu$41 outside $\mu$42 from the values of $\mu$43 and $\mu$44 on $\mu$45, with coefficients expressed by theta-function Wronskians (Casper et al., 2024).

An alternative recent approach replaces the local Wirtinger–Poincaré step in KNS by ground-state estimates for confined harmonic oscillators. In the Fourier-symmetric Sobolev space

$\mu$46

the uniqueness threshold is again controlled by the product of interval lengths and radii, now with the sharp constant $\mu$47 arising from the Dirichlet ground-state energy of the confined oscillator (Szehr, 18 Sep 2025).

5. Phaseless and nonlinear variants: discrete Pauli pairs

The Pauli problem replaces linear vanishing by modulus constraints. Two functions $\mu$48 form a classical Pauli pair when

$\mu$49

A discrete Pauli pair consists of two discrete sets $\mu$50 such that

$\mu$51

This is a nonlinear analogue of a Fourier uniqueness pair (Ramos et al., 2024).

A first theorem shows that discrete modulus equalities can propagate to global modulus equalities under Gaussian decay and critical accumulation. If $\mu$52, one of them has simultaneous Gaussian decay in space and frequency, and

$\mu$53

then

$\mu$54

on all of $\mu$55. Sequences of the form $\mu$56 are the model case. The proof propagates Gaussian decay from the sampled nodes to the whole line, then uses entire-function continuation and zero-density estimates of order $\mu$57 (Ramos et al., 2024).

The same paper proves that both hypotheses are essential. Without decay, for arbitrary discrete $\mu$58 there exist smooth $\mu$59 matching the sampled moduli on $\mu$60 and $\mu$61 but not globally. In the supercritical regime

$\mu$62

there is an infinite-dimensional family of Schwartz counterexamples with Gaussian bounds. As a by-product, the same machinery yields a sharp discrete Hardy uncertainty principle: if $\mu$63, sufficiently dense discrete Gaussian bounds on $\mu$64 and $\mu$65 force $\mu$66 (Ramos et al., 2024).

A later paper identifies the sharp density thresholds in the symmetric Hardy class

$\mu$67

For one-sided modulus uniqueness, the critical constant is

$\mu$68

If

$\mu$69

then sampled equality $\mu$70 on $\mu$71 forces $\mu$72 globally; if the liminf exceeds $\mu$73, the conclusion fails (Lysen, 21 May 2026).

For full two-sided discrete Pauli pairs, the sharp threshold becomes $\mu$74. There is also a sharp weak-Pauli threshold

$\mu$75

Below this density, a discrete Pauli pair must be a weak Pauli pair; above it, there are counterexamples in which neither the time-side nor the frequency-side modulus agrees globally (Lysen, 21 May 2026).

A persistent misconception is that these results solve phase retrieval. They do not. The conclusions are global equality of moduli, or weak Pauli equivalence, not the stronger statement $\mu$76. This limitation is stated explicitly in the Pauli-pair literature (Ramos et al., 2024).

6. Group-valued transforms, multiplier analogues, and open directions

The Euclidean theory has noncommutative analogues. For the symplectic Fourier transform on $\mu$77, the pair $\mu$78 is a Heisenberg uniqueness pair if and only if the complex cone $\mu$79 is non-harmonic. The same work shows that spheres and non-harmonic cones are determining sets for the special Hermite spectral projections of finite measures supported on the sphere, and proves a finite-rank theorem on step-two nilpotent Lie groups: if the operator-valued Fourier transform has arbitrary finite rank, the function must vanish (Ghosh et al., 2018).

On the Euclidean motion group $\mu$80, operator-valued Fourier transforms generate another variant. If $\mu$81 and $\mu$82 has finite rank for every $\mu$83 and fixed $\mu$84, then $\mu$85. The same paper formulates HUPs on $\mu$86 and on $\mu$87, and proves that $\mu$88 is a HUP in $\mu$89 if and only if $\mu$90 is a HUP in $\mu$91 (Chattopadhyay et al., 2016).

The uniqueness-pair paradigm also extends beyond the Fourier transform itself. For the fractional Laplacian $\mu$92, $\mu$93, a pair of discrete sets $\mu$94 is a uniqueness pair in a function class $\mu$95 if

$\mu$96

Using transformed Beurling densities defined through the maps

$\mu$97

the theory proves sufficient uniqueness conditions in Schwartz space, non-uniqueness for broad sparse regimes when $\mu$98, and a lattice non-uniqueness theorem in all dimensions: $\mu$99 for every $S$00 and every $S$01. The framework further extends to general multiplier operators $S$02, with uniqueness under an analytic barrier and lattice non-uniqueness under a symbol-difference condition (Motta, 14 Apr 2026).

Several open directions recur across the literature. For quadratic hypersurfaces, the characterization of the exceptional set $S$03, extensions beyond finite measures, and general algebraic varieties remain open (Gröchenig et al., 2016). For phaseless sampling, optimal constants in the Gaussian regime, the borderline Hardy case $S$04, and higher-dimensional point-sampling analogues are explicitly posed (Ramos et al., 2024, Lysen, 21 May 2026). For the fractional Laplacian, extending the one-dimensional $S$05 non-uniqueness theory to general $S$06 is open (Motta, 14 Apr 2026). A plausible overarching implication is that critical-density geometry, rather than any single transform-specific trick, is the organizing principle across linear, phaseless, and multiplier versions of Fourier uniqueness.