Critical Uniqueness Pairs in Fourier Analysis

• Critical Uniqueness Pairs are threshold configurations in Fourier uniqueness theory that mark the transition between uniqueness and nonuniqueness with finite-dimensional defects.
• They are exemplified by the hyperbola/Klein–Gordon model, where lattice-cross sampling creates a sharp spacing threshold resulting in a defect of 1.
• Dynamical reformulations using Gauss-type maps and oscillatory integrals reveal rigid zero geometries, such as the Nielsen spiral, that characterize these critical cases.

Searching arXiv for the main paper and closely related work on Fourier/Heisenberg uniqueness pairs. Critical uniqueness pairs arise in Fourier uniqueness theory at the exact transition between uniqueness and nonuniqueness. In Hedenmalm’s reformulation of Heisenberg uniqueness pairs as Fourier uniqueness sets, the decisive phenomenon is often a sharp spacing threshold on the Fourier side, together with a possible finite-dimensional defect at equality. The paradigmatic example is the hyperbola associated with the one-dimensional Klein–Gordon equation, tested against a lattice-cross in the Fourier plane: for one branch of the hyperbola the threshold is exact, the critical case has defect $1$, and the corresponding nontrivial solution exhibits rigid zero geometry explained by the Nielsen spiral (Hedenmalm, 2011).

1. Terminology and conceptual framework

Let $\Gamma\subset \mathbb R^n$ be a prescribed support set and let $\mu$ be a finite complex Borel measure supported on $\Gamma$. Its Fourier transform is

$$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$

If $\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$ is a linear space of measures supported on $\Gamma$, then $\Lambda\subset\mathbb R^n$ is a Fourier uniqueness set for $\mathrm X(\Gamma)$ if

$$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$

Equivalently, $\Gamma\subset \mathbb R^n$0 is a Fourier uniqueness pair for $\Gamma\subset \mathbb R^n$1. In the classical terminology of Hedenmalm–Montes-Rodríguez, when $\Gamma\subset \mathbb R^n$2, the absolutely continuous measures on $\Gamma\subset \mathbb R^n$3, $\Gamma\subset \mathbb R^n$4 is called a Heisenberg uniqueness pair (Hedenmalm, 2011).

A decisive generalization is the notion of defect. A set $\Gamma\subset \mathbb R^n$5 is a Fourier uniqueness set of defect $\Gamma\subset \mathbb R^n$6 for $\Gamma\subset \mathbb R^n$7 if the solution space

$\Gamma\subset \mathbb R^n$8

has dimension $\Gamma\subset \mathbb R^n$9. Defect $\mu$0 is uniqueness. The paper also introduces a local version: for $\mu$1, one says $\mu$2 is a $\mu$3-local Fourier uniqueness set if

$\mu$4

This framework enlarges the older HUP theory from absolutely continuous measures to all finite Borel measures $\mu$5, to the Hardy-type space $\mu$6, and to one-branch restrictions such as $\mu$7 (Hedenmalm, 2011).

A plausible implication is that a “critical uniqueness pair” is best understood not as an independent formal definition but as a threshold configuration in which the uniqueness relation is exact and the corresponding annihilator is either trivial, finite-dimensional, or newly nontrivial. In the hyperbola/Klein–Gordon model, this interpretation is explicit through the defect formalism and the sharp equality cases (Hedenmalm, 2011).

2. Hyperbolic geometry and the Klein–Gordon correspondence

The central geometric object is

$\mu$8

with branches

$\mu$9

A convenient parametrization of the positive branch is

$\Gamma$0

The canonical testing set is the lattice-cross

$\Gamma$1

It samples the two coordinate axes at spacings $\Gamma$2 and $\Gamma$3 (Hedenmalm, 2011).

The analytic meaning of this geometry comes from the one-dimensional Klein–Gordon equation

$\Gamma$4

In null coordinates

$\Gamma$5

it becomes

$\Gamma$6

If

$\Gamma$7

then $\Gamma$8 solves this equation if and only if

$\Gamma$9

hence

$$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$0

Thus the hyperbola is the characteristic variety in the $$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$1-variables, and vanishing of $$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$2 on $$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$3 becomes a unique continuation problem for Klein–Gordon solutions of Fourier-transform type (Hedenmalm, 2011).

The full-hyperbola threshold, established in earlier work and taken as the starting point of the 2011 paper, is

$$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$4

The one-branch problem is subtler and is the source of the critical phenomenon that dominates later discussion (Hedenmalm, 2011).

3. Sharp critical threshold for one branch

For the restriction to a single branch, the paper proves the sharp theorem

$$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$5

and at the critical value

$$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$6

the pair is a HUP with defect $$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$7. This is the exact critical threshold for the one-branch problem (Hedenmalm, 2011).

The term “critical density” is used in the paper in the sense of the lattice-spacing product $$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$8, scaled by $$\widehat{\mu}(\xi)=\int_{\Gamma} e^{-2\pi i x\cdot \xi}\,d\mu(x), \qquad \xi\in\mathbb{R}^n.$$9, rather than in a Beurling-density sense. The critical relation is

$\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$0

for $\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$1, whereas for the full hyperbola it is

$\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$2

Restricting to one branch therefore raises the critical threshold by a factor of $\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$3 (Hedenmalm, 2011).

This distinction is one of the paper’s most striking structural findings. The full hyperbola and one branch share the same characteristic equation, yet the uniqueness threshold changes dramatically. This suggests that the missing branch carries a genuine rigidity contribution, not merely a redundancy.

Model	Testing set	Sharp threshold
Full hyperbola $\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$4	$\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$5	HUP iff $\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$6
One branch $\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$7	$\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$8	HUP iff $\mathrm X(\Gamma)\subset \mathrm M(\Gamma)$9
One branch, critical case	$\Gamma$0	defect $\Gamma$1 when $\Gamma$2

The proof mechanism passes through a dynamical reformulation. After scaling to

$\Gamma$3

and writing $\Gamma$4, the annihilation conditions reduce to

$\Gamma$5

These are transformed into periodization relations governed by the Gauss-type map

$\Gamma$6

If $\Gamma$7, iterates of $\Gamma$8 force all mass to disappear; if $\Gamma$9, the invariant measure for the standard Gauss map survives and gives defect $\Lambda\subset\mathbb R^n$0; if $\Lambda\subset\mathbb R^n$1, there are nontrivial invariant densities and hence nonuniqueness. Since $\Lambda\subset\mathbb R^n$2 corresponds exactly to $\Lambda\subset\mathbb R^n$3, the critical threshold is encoded by a precise dynamical transition (Hedenmalm, 2011).

4. Critical defect, rigidity, and the Nielsen spiral

At the critical value $\Lambda\subset\mathbb R^n$4, the annihilator is one-dimensional. In normalized form $\Lambda\subset\mathbb R^n$5, $\Lambda\subset\mathbb R^n$6, a spanning nontrivial measure on $\Lambda\subset\mathbb R^n$7 is

$\Lambda\subset\mathbb R^n$8

Its span is exactly the defect space (Hedenmalm, 2011).

The corresponding Fourier transform vanishes on the entire lattice-cross,

$\Lambda\subset\mathbb R^n$9

but is not identically zero. The critical nonuniqueness is also maximally rigid: if $\mathrm X(\Gamma)$0 is a single extra point on the coordinate cross but not already in $\mathrm X(\Gamma)$1, then

$\mathrm X(\Gamma)$2

Thus one additional off-lattice point on the cross removes the defect (Hedenmalm, 2011).

A stronger statement follows. If

$\mathrm X(\Gamma)$3

then the nontrivial critical solution cannot vanish at $\mathrm X(\Gamma)$4. Equivalently,

$\mathrm X(\Gamma)$5

for every nonzero axis point outside the lattice-cross. The only zeros of the critical solution on the axes are the prescribed lattice points (Hedenmalm, 2011).

The mechanism behind this rigidity is an explicit oscillatory integral. In normalized variables,

$\mathrm X(\Gamma)$6

satisfies

$\mathrm X(\Gamma)$7

To exclude additional zeros, the paper reduces to

$\mathrm X(\Gamma)$8

The associated curve

$\mathrm X(\Gamma)$9

is the Nielsen spiral. It spirals into the origin as $$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$0, but never passes through the origin. Therefore the oscillatory integral never vanishes, and the only zeros come from the prefactor $$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$1, namely at integers (Hedenmalm, 2011).

This is the paper’s most unexpected geometric ingredient: critical nonuniqueness survives in exactly one direction, yet the surviving solution has a zero set on the axes that is completely rigid.

5. Analytic structure, locality, and proof machinery

The one-branch theorem sits inside a larger reorganization of Fourier uniqueness theory. The paper distinguishes strong HUPs, meaning uniqueness in $$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$2, from weak HUPs, meaning uniqueness in $$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$3, and develops local Fourier uniqueness sets. This leads to results that go beyond the original absolute-continuity setting (Hedenmalm, 2011).

A major tool is the Hilbert transform on the hyperbola. For $$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$4, so that $$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$5, the natural symmetric measure is

$$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$6

and for a measure $$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$7 on $$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$8,

$$\forall \mu\in \mathrm X(\Gamma):\qquad \widehat{\mu}\big|_{\Lambda}=0 \implies \mu=0.$$9

Under compression to the $\Gamma\subset \mathbb R^n$00-axis,

$\Gamma\subset \mathbb R^n$01

This permits the definition of the Hardy-like Banach space $\Gamma\subset \mathbb R^n$02, and the transform identities

$\Gamma\subset \mathbb R^n$03

which are essential in the local theory (Hedenmalm, 2011).

The paper also proves strong and local statements for the full hyperbola. If $\Gamma\subset \mathbb R^n$04 is a Riesz set for $\Gamma\subset \mathbb R^n$05, then

$\Gamma\subset \mathbb R^n$06

is a Fourier uniqueness set for $\Gamma\subset \mathbb R^n$07 if and only if

$\Gamma\subset \mathbb R^n$08

For open quadrants $\Gamma\subset \mathbb R^n$09, there is a sharp time-like/space-like split: if $\Gamma\subset \mathbb R^n$10, then $\Gamma\subset \mathbb R^n$11 is not a local uniqueness set for $\Gamma\subset \mathbb R^n$12; if $\Gamma\subset \mathbb R^n$13, then $\Gamma\subset \mathbb R^n$14 is a $\Gamma\subset \mathbb R^n$15-local Fourier uniqueness set for $\Gamma\subset \mathbb R^n$16 if and only if

$\Gamma\subset \mathbb R^n$17

The paper also analyzes distorted lattice-crosses and obtains refined weak-HUP criteria involving the shift parameter $\Gamma\subset \mathbb R^n$18 (Hedenmalm, 2011).

Methodologically, the proofs combine harmonic analysis, Hardy space duality, distribution theory, ergodic properties of Gauss-type maps, complex analysis, and pseudocontinuation. A plausible implication is that critical uniqueness pairs in this setting are not controlled by geometry alone: they are simultaneously dynamical, Hardy-theoretic, and PDE-theoretic objects.

6. Later developments and broader critical regimes

Subsequent work extends the critical-pair perspective in several directions. Jaming and Kellay recast HUPs for curves and unions of lines through the origin as a dynamical system on the curve generated by projection involutions. Their method yields general criteria based on wandering intervals, attractive intervals, and irrational rotations, and unifies known cases such as circle, parabola, and hyperbola (Jaming et al., 2013). This dynamical viewpoint is consonant with the Gauss-map mechanism already visible in the one-branch hyperbola problem (Hedenmalm, 2011).

Huang and Yu study the endpoint $\Gamma\subset \mathbb R^n$19, corresponding to the wave equation rather than Klein–Gordon. For $\Gamma\subset \mathbb R^n$20, the Fourier transform takes the form

$\Gamma\subset \mathbb R^n$21

and uniqueness for vanishing on two curves is characterized by density of projections together with a dynamical wandering-set condition for the map $\Gamma\subset \mathbb R^n$22. This shows that the endpoint case is qualitatively different from the massive hyperbola/Klein–Gordon regime (Huang et al., 2023).

On the discrete side, Kulikov, Nazarov, and Sodin formulate Fourier uniqueness pairs for discrete subsets of $\Gamma\subset \mathbb R^n$23 by simultaneous zero conditions

$\Gamma\subset \mathbb R^n$24

They identify supercritical and subcritical regimes through weighted gap conditions with threshold $\Gamma\subset \mathbb R^n$25, proving uniqueness in the supercritical case and nonuniqueness in the subcritical case (Kulikov et al., 2023). Later work constructs sufficient conditions exactly at the critical threshold $\Gamma\subset \mathbb R^n$26, including highly asymmetric pairs and families close to the Shannon–Whittaker extreme; these are explicitly described as critical uniqueness pairs for the Fourier-symmetric Sobolev space $\Gamma\subset \mathbb R^n$27 (Lysen, 22 Sep 2025). In higher dimensions, an analogous density transition occurs at

$\Gamma\subset \mathbb R^n$28

with sparse critical pairs yielding infinite-dimensional kernels and dense critical pseudohomogeneous lattice images yielding injectivity for Schwartz functions (Adve, 2023).

A further development comes from the Radchenko–Viazovska interpolation setting. The basis functions of the Fourier interpolation formula possess many extraneous zeros, and those zeros give Fourier nonuniqueness pairs $\Gamma\subset \mathbb R^n$29 for even Schwartz functions, despite an apparent excess of sampling points over the canonical uniqueness set $\Gamma\subset \mathbb R^n$30 (Berghaus et al., 21 Dec 2025). This suggests that criticality in Fourier uniqueness theory is not exhausted by gap asymptotics; intrinsic zero geometry of the basis functions can itself generate threshold failures.

Across these works, the recurrent pattern is sharp transition. In one branch of the hyperbola the transition is

$\Gamma\subset \mathbb R^n$31

with defect $\Gamma\subset \mathbb R^n$32 (Hedenmalm, 2011). In discrete Sobolev-space problems it is the weighted local-spacing threshold $\Gamma\subset \mathbb R^n$33 (Kulikov et al., 2023, Lysen, 22 Sep 2025). In higher-dimensional discrete density theory it is $\Gamma\subset \mathbb R^n$34 (Adve, 2023). What persists is the same structural phenomenon: the critical pair is the place where uniqueness ceases to be a purely yes-or-no property and acquires finite-dimensional defects, dynamical obstructions, or explicit nonuniqueness modes.