--- title: Fourier Uniqueness Pairs url: https://www.emergentmind.com/topics/fourier-uniqueness-pairs type: topic --- # Fourier Uniqueness Pairs 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 [1608.06738]. ## 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 \(S\) have Fourier transforms agreeing on \(\Lambda\), then the measures are equal [1608.06738]. A parallel function-space formulation is standard in the discrete theory. For a function class \(H\) on \(\mathbb{R}\), a pair \((\Lambda,\Gamma)\subset\mathbb{R}\times\mathbb{R}\) is a uniqueness pair if \[ f|_{\Lambda}=0,\qquad \widehat f|_{\Gamma}=0 \quad\Longrightarrow\quad f\equiv 0 \] for all \(f\in H\); 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 [2306.14013]. 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 \(1\), \(1/2\), \(2\), \(\pi\), or \(4\pi^2\) in different articles: the underlying uniqueness question is invariant, but the constants rescale with the chosen convention [2605.23079]. 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 [1608.06738]. ## 2. Planar archetypes: hyperbolas, circles, parabolas, and dynamical systems The foundational planar example is the hyperbola \[ \Gamma_{\varepsilon}=\{(x_1,x_2)\in\mathbb{R}^2:\ x_1x_2=\varepsilon\},\qquad \varepsilon\neq 0, \] paired with the lattice cross \[ \Lambda_{\alpha\beta}=(\alpha\mathbb{Z}\times\{0\})\cup(\{0\}\times \beta\mathbb{Z}). \] Hedenmalm–Montes-Rodríguez proved that \((\Gamma_{\varepsilon},\Lambda_{\alpha\beta})\) is a HUP if and only if \(\alpha\beta\le 1/|\varepsilon|\). For the one-branch hyperbola, the threshold changes: \((\Gamma_m^+,\Lambda_{\alpha,\beta})\) is HUP if and only if \(\alpha\beta m^2<16\pi^2\), while at \(\alpha\beta m^2=16\pi^2\) the defect is one-dimensional [2311.15601; 1106.2985]. The Klein–Gordon connection is intrinsic. If \(u=\widehat{\mu}\) with \(\mu\) supported on \(\Gamma_m\), then \(u\) solves \[ \partial_{\xi_1}\partial_{\xi_2}u=\tfrac14 m^2 u. \] 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 [1106.2985]. A major conceptual advance is the dynamical-system reformulation of vanishing on lines. When \(\Lambda\) is a union of lines through the origin, vanishing of \(\widehat{\mu}\) on a line can be rewritten as a weighted antisymmetry relation across the fibers of the corresponding projection. For two lines, the resulting involutions \(\Phi_{\theta_1}\) and \(\Phi_{\theta_2}\) generate a map \(\Phi=\Phi_{\theta_2}\circ\Phi_{\theta_1}\) 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 \(\mathbb{Q}\pi\); it also unifies proofs for circles, parabolas, and hyperbolas and produces new examples for cusps, polygons, and smooth convex curves [1312.6236]. The endpoint case of the hyperbola family, namely \[ \Gamma_0=\{(x_1,x_2)\in\mathbb{R}^2:\ x_1x_2=0\}, \] is qualitatively different. Here \(\Gamma_0\) is the union of the coordinate axes, and any \(\mu\in AC(\Gamma_0)\) decomposes as \[ d\mu(x_1,x_2)=f(x_1)\,dx_1\otimes\delta_0(dx_2)+g(x_2)\,\delta_0(dx_1)\otimes dx_2, \] so that \[ \widehat\mu(\xi_1,\xi_2)=\widehat f(\xi_1)+\widehat g(\xi_2). \] For this wave-equation endpoint, density of the coordinate projections of \(\Lambda\) is necessary, but in general not sufficient. When \(\Lambda\) lies in the union of two graphs \(A_1,A_2\) of continuous bijections \(T_1,T_2\), the decisive object is \(\Phi=T_2^{-1}\circ T_1\): under a mild nondegeneracy hypothesis, \((\Gamma_0,A_1\cup A_2)\) is HUP if and only if \(\pi_1(A_1\cup A_2)\) is dense and the open set \(\mathbb{R}\setminus(\pi_1(A_1)\cup\pi_1(A_2))\) is wandering for \(\Phi\) [2311.15601]. ## 3. Quadratic hypersurfaces, cones, and higher-dimensional geometry A higher-dimensional turning point is the quadratic-hypersurface theorem of Gröchenig and Jaming. Let \[ S=\{x\in\mathbb{R}^d:\ Q(x)+2\langle v,x\rangle=p\}, \] where \(Q(x)=B(x,x)\) is a quadratic form. Then there exists an exceptional set \(E(Q,v,p)\subset S^{d-1}\times S^{d-1}\) of measure zero such that, if \(u_1,u_2\in S^{d-1}\) are distinct, satisfy \(Q(u_1)\neq 0\), \(Q(u_2)\neq 0\), and \((u_1,u_2)\notin E(Q,v,p)\), the pair \[ \bigl(S,\ H_{u_1}\cup H_{u_2}\bigr) \] 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 [1608.06738]. The proof rests on three geometric mechanisms. First, vanishing on a single hyperplane with non-isotropic normal \(u\) yields an odd symmetry under the affine involution \[ R_{u,v}(x)=x-2\,\frac{B(x,u)+\langle v,u\rangle}{Q(u)}\,u. \] 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 [1608.06738]. A sharp corollary is a Cramér–Wold theorem specialized to quadratic supports: a finite measure supported on \(S\) is uniquely determined by its projections onto two generic hyperplanes, whereas an arbitrary finite measure on \(\mathbb{R}^d\) requires a dense family of projections. The same reflection mechanism also yields a unique-continuation statement for second-order constant-coefficient PDEs: if \(P(D)\mu=0\) and \(\mu\) vanishes on two generic hyperplanes, then \(\mu\equiv 0\) [1608.06738]. The theorem is not a generic statement about arbitrary supports. For a single hyperplane \(H_{u,s}\), \((S,H_{u,s})\) is a HUP if and only if the orthogonal projection \(T_u\) is one-to-one on \(S\). In particular, if \(Q(u)\neq 0\), then \((S,H_{u,s})\) is generally not a HUP; if \(Q(u)=0\), uniqueness depends on the isotropic set \[ E_u=\{x\in S:\ B(x,u)=-\langle v,u\rangle\}. \] On spheres, the parallel-hyperplane case is especially sharp: for general finite measures on \(S^{d-1}\), \((S^{d-1},H_{u,s_1}\cup H_{u,s_2})\) is a HUP if and only if \(|s_1-s_2|>2\), while for absolutely continuous measures it is an \(AC(S^{d-1})\)-HUP for any \(s_1\neq s_2\). If finitely many hyperplanes through the origin are used, the exact criterion is Coxeter-theoretic: \((S^{d-1},H_1\cup\cdots\cup H_N)\) is a HUP if and only if the Coxeter group generated by the orthogonal reflections in the \(H_k\) is infinite [1608.06738]. A complementary geometric theorem concerns cones. For the Euclidean Fourier transform on \(\mathbb{R}^n\), \((S^{n-1},C)\) is a HUP if and only if the cone \(C\) 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 \(\mathbb{R}^3\), the paraboloid and certain circles on \(S_R(o)\) are HUPs for a specific symmetric class of finite Borel measures on \(S^2\) [1507.02624]. ## 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 \(\Lambda=(\lambda_j)\) and \(\Gamma=(\mu_j)\) with \(11\). Any supercritical pair is a uniqueness pair for the Schwartz space, indeed for the Fourier-symmetric Sobolev space \(H=\{f:\ f,\widehat f\in H^1(\mathbb{R})\}\), while any subcritical pair is a non-uniqueness pair already in a Gelfand–Shilov space \(\mathcal{S}(p,q)\) [2306.14013]. In the same setting, separated supercritical pairs satisfy a two-sided frame inequality in \(H_{s,p,q}\) and admit a Fourier interpolation formula \[ f(x)=\sum_{\lambda\in\Lambda} f(\lambda)\,a_\lambda(x)+\sum_{\mu\in\Gamma}\widehat f(\mu)\,b_\mu(x), \] with convergence in \(H_{s,p,q}\). This interpolation, in turn, produces locally finite measures with discrete support and discrete Fourier transform, giving an explicit source of crystalline measures [2306.14013]. The model family \[ \lambda_j=\operatorname{sign}(j)\,(\alpha |j|)^{1/p},\qquad \mu_j=\operatorname{sign}(j)\,(\alpha |j|)^{1/q} \] exhibits the threshold sharply: \(\alpha<1\) is supercritical, \(\alpha>1\) is subcritical, and the case \(p=q=2,\alpha=1\) is the critical square-root scale associated with Radchenko–Viazovska interpolation [2306.14013]. The critical regime itself is subtler. A recent sufficient condition uses a subexponentially admissible weight \(S\) and shifted gap controls \[ \sup_j S(|\lambda_j|+\tilde d)\,(\lambda_{j+1}-\lambda_j)\le \tfrac12,\qquad \sup_j S^{-1}(|\mu_j|+d)\,(\mu_{j+1}-\mu_j)\le \tfrac12, \] and yields Fourier uniqueness pairs in the critical case. This produces strongly asymmetric examples such as \[ \lambda_j \sim \left(\frac{\log j}{4\pi}\right)^{1/s},\qquad \mu_j \sim \frac{j}{2}\left(\frac{4\pi}{\log j}\right)^{1/s}, \] which approach the Bondarenko–Radchenko–Seip profile and move toward the asymmetry of Shannon–Whittaker sampling while remaining purely discrete on both sides [2509.17600]. In higher dimensions, density can be expressed through variable-radius coverings. For closed discrete \(A,B\subset\mathbb{R}^d\), the notions of \((p,\delta)\)-dense and \((p,\delta)\)-separated control whether the map \[ F_{A,B}(f)=\bigl(f|_A,\widehat f|_B\bigr) \] is injective or surjective on \(\mathcal{S}(\mathbb{R}^d)\). If \(pq>1\) and both sets are sufficiently dense, injectivity holds for small \(\delta\); if \(pq<1\) and both are sufficiently separated, the kernel is infinite-dimensional and, for large \(\delta\), arbitrary rapidly decreasing data can be prescribed on \(A\) and \(B\). At criticality \(pq=1\), small-\(\delta\) injective examples arise from pseudohomogeneous deformations of lattices, while large-\(\delta\) separated sets yield infinite-dimensional kernels. A concrete example is \[ A=\{\delta |n|^{t-1}n:\ n\in\mathbb{Z}^d\},\qquad B=\{\delta |n|^{u-1}n:\ n\in\mathbb{Z}^d\},\qquad t+u=1, \] which is a Fourier uniqueness pair for sufficiently small \(\delta\) and a non-uniqueness pair for sufficiently large \(\delta\) [2306.07475]. Arithmetic specializations sharpen this picture. For \(0<\alpha,\beta<1\), if \(f\in\mathcal{S}(\mathbb{R})\) vanishes on \(\pm n^\alpha\) and \(\widehat f\) vanishes on \(\pm n^\beta\), uniqueness is proved on an explicit region \[ A=\Bigl\{(\alpha,\beta)\in[0,1]^2:\ \alpha+\beta<1,\ \text{and either } \alpha<1-\frac{\beta}{1-\alpha-\beta}\ \text{or}\ \beta<1-\frac{\alpha}{1-\alpha-\beta}\Bigr\}, \] with the diagonal corollary \(\alpha<1-\frac{\sqrt2}{2}\) [1910.04276]. A plausible implication, explicitly conjectured there, is that the full subcritical condition \(\alpha+\beta<1\) should be sufficient. Critical symmetry does not preclude non-uniqueness. The basis functions underlying the Radchenko–Viazovska interpolation formula on \(\{\sqrt n\}\) 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 [2512.18677]. A finite cyclic analogue also exists. For the discrete Fourier transform on \(L^2(\mathbb{Z}/N\mathbb{Z})\), the restricted space \[ \mathcal C_a=\{f:\ \operatorname{supp}(f)\subseteq [-a,a],\ \operatorname{supp}(\widehat f)\subseteq [-a,a]\} \] is governed by a commuting periodically tridiagonal matrix \(J\). The simple eigenspaces of \(J\) span \(\mathcal C_a\), and the dimension formula is \[ \dim \mathcal C_a = 4a+2-N \quad \text{for } a\ge \frac{N-1}{4}. \] For \((N-2)/4\le a\le (N-1)/2\), there is an explicit interpolation formula reconstructing \(f(x)\) outside \([-a,a]\) from the values of \(f\) and \(\widehat f\) on \([-a,a]\), with coefficients expressed by theta-function Wronskians [2407.20379]. 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 \[ \mathcal H=\Bigl\{f\in L^2(\mathbb R): \int_{\mathbb R}\bigl(x^2|f(x)|^2+|f'(x)|^2\bigr)\,dx<\infty\Bigr\}, \] the uniqueness threshold is again controlled by the product of interval lengths and radii, now with the sharp constant \(\pi\) arising from the Dirichlet ground-state energy of the confined oscillator [2509.14953]. ## 5. Phaseless and nonlinear variants: discrete Pauli pairs The Pauli problem replaces linear vanishing by modulus constraints. Two functions \(f,g:\mathbb{R}\to\mathbb{C}\) form a classical Pauli pair when \[ |f(x)|=|g(x)| \quad\text{for a.e. }x,\qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{for a.e. }\xi. \] A discrete Pauli pair consists of two discrete sets \(\Lambda,\Gamma\subset\mathbb{R}\) such that \[ |f(\lambda)|=|g(\lambda)|\quad(\lambda\in\Lambda),\qquad |\widehat f(\gamma)|=|\widehat g(\gamma)|\quad(\gamma\in\Gamma). \] This is a nonlinear analogue of a Fourier uniqueness pair [2410.12065]. A first theorem shows that discrete modulus equalities can propagate to global modulus equalities under Gaussian decay and critical accumulation. If \(f,g\in H^1\cap \mathcal F(H^1)\), one of them has simultaneous Gaussian decay in space and frequency, and \[ \max\Big\{\limsup |\lambda_{i+1}-\lambda_i|\,|\lambda_i|,\ \limsup |\gamma_{i+1}-\gamma_i|\,|\gamma_i|\Big\}<\frac1{C_\alpha}, \] then \[ |f(x)|\equiv |g(x)|,\qquad |\widehat f(\xi)|\equiv |\widehat g(\xi)| \] on all of \(\mathbb{R}\). Sequences of the form \(\pm c\sqrt n\) 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 \(2\) [2410.12065]. The same paper proves that both hypotheses are essential. Without decay, for arbitrary discrete \(\Lambda,\Gamma\) there exist smooth \(f,g\) matching the sampled moduli on \(\Lambda\) and \(\Gamma\) but not globally. In the supercritical regime \[ \min\Big\{\liminf |\lambda_{i+1}-\lambda_i|\,|\lambda_i|,\ \liminf |\gamma_{i+1}-\gamma_i|\,|\gamma_i|\Big\}>\frac12, \] 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 \(A>1\), sufficiently dense discrete Gaussian bounds on \(f\) and \(\widehat f\) force \(f\equiv 0\) [2410.12065]. A later paper identifies the sharp density thresholds in the symmetric Hardy class \[ E(A,A)=\Bigl\{f:\ \sup_x |f(x)|e^{A\pi |x|^2}+\sup_\xi |\widehat f(\xi)|e^{A\pi |\xi|^2}<\infty\Bigr\},\qquad 00\) and fixed \(\sigma\), then \(f\equiv 0\). The same paper formulates HUPs on \(M(n)\) and on \(\mathbb{R}^n\times K\), and proves that \((T,A)\) is a HUP in \(\mathbb{R}^n\) if and only if \((T\times K, A\times K)\) is a HUP in \(\mathbb{R}^n\times K\) [1608.00825]. The uniqueness-pair paradigm also extends beyond the Fourier transform itself. For the fractional Laplacian \(T=(-\Delta)^s\), \(00\). The framework further extends to general multiplier operators \(T_m\), with uniqueness under an analytic barrier and lattice non-uniqueness under a symbol-difference condition [2604.12997]. Several open directions recur across the literature. For quadratic hypersurfaces, the characterization of the exceptional set \(E(Q,v,p)\), extensions beyond finite measures, and general algebraic varieties remain open [1608.06738]. For phaseless sampling, optimal constants in the Gaussian regime, the borderline Hardy case \(A=1\), and higher-dimensional point-sampling analogues are explicitly posed [2410.12065; 2605.23079]. For the fractional Laplacian, extending the one-dimensional \(s=1/2\) non-uniqueness theory to general \(s\in(0,1)\) is open [2604.12997]. 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. Source: https://www.emergentmind.com/topics/fourier-uniqueness-pairs