[Paper Review] Uniqueness Theorems for Fourier Quasicrystals and Temperate Distributions with Discrete Support

S. Favorov|arXiv (Cornell University)|Jun 13, 2020

Analytic and geometric function theory4 citations

TL;DR

This paper establishes uniqueness theorems for Fourier quasicrystals and discrete temperate distributions by showing that if two such distributions have increasingly close supports and matching masses at corresponding points at infinity, they must be identical. The key result is that asymptotic proximity of masses and locations in the support implies full equality, even under weakened sparsity and localization conditions on a set with specific geometric density.

ABSTRACT

It is proved that if some points of the supports of two Fourier quasicrystals approach each other while tending to infinity and the same is true for the masses at these points, then these quasicrystals coincide. A similar statement is obtained for a certain class of discrete temperate distributions.

Motivation & Objective

• To establish conditions under which two Fourier quasicrystals or discrete temperate distributions must be identical based on asymptotic behavior of their supports and masses.
• To generalize previous results on almost periodic functions and zero sets to broader classes of distributions with discrete support and spectrum.
• To weaken the assumptions of sparsity and uniform discreteness while preserving uniqueness under asymptotic proximity conditions.
• To extend the theory to distributions that are not necessarily measures but atomic or tempered distributions with discrete structure.
• To investigate whether the uniqueness principle holds for almost periodic measures beyond the atomic case, particularly when components are mixed with absolutely continuous parts.

Proposed method

• Define sparse Fourier quasicrystals as discrete temperate distributions whose Fourier transforms are atomic and whose total variation measures are also temperate.
• Use the Fourier transform on tempered distributions via duality: ⟨f̂, φ⟩ = ⟨f, φ̂⟩ for φ ∈ S(ℝ^d).
• Introduce a geometric condition (4) on a set E with balls B(x_k, r_k) where r_k → ∞ and r_k/|x_k| → 0, ensuring E is sufficiently dense at infinity.
• Apply almost periodicity of translation functions H(t) = f(ψ_t) − g(ψ_t) derived from test functions ψ_t with compactly supported derivatives.
• Use contradiction via uniform convergence of distributional coefficients: if H(t) ≠ 0 at a point but |H(t)| → 0 along a relatively dense set, contradiction arises.
• Leverage the fact that derivatives of test functions are uniformly continuous and bounded, allowing approximation of distributional values at distant points by local behavior.

Experimental results

Research questions

• RQ1 Under what conditions do two Fourier quasicrystals with asymptotically close supports and masses necessarily coincide?
• RQ2 Can the uniqueness result be extended beyond uniformly discrete or sparse measures to more general discrete temperate distributions?
• RQ3 Does the coincidence of two distributions follow if their supports and masses approach each other along a set E satisfying condition (4), even without global sparsity?
• RQ4 Can the uniqueness principle be extended to distributions with both atomic and absolutely continuous components?
• RQ5 Is the uniqueness result valid for general almost periodic measures, not just those with discrete spectrum and support?

Key findings

• If two sparse Fourier quasicrystals μ and ν satisfy λ_n − γ_n → 0 and μ(λ_n) − ν(γ_n) → 0 as n → ∞, then μ ≡ ν.
• The uniqueness condition can be localized: if the supports and masses are asymptotically close on a set E with property (4), then μ ≡ ν, even without global sparsity.
• The result extends to grouped supports: if μ and ν are atomic and their components Λ_n, Γ_n are grouped such that diam(Λ_n ∪ Γ_n) → 0 and μ(Λ_n) − ν(Γ_n) → 0, then μ ≡ ν.
• The proof relies on constructing an almost periodic function H(t) = f(ψ_t) − g(ψ_t) that is non-zero at a point but tends to zero along a relatively dense set, leading to contradiction.
• The method applies to general discrete temperate distributions, not just measures, by analyzing the behavior of distributional coefficients under translation and localization.
• The result holds even when the total variation of the measure is not in S*(ℝ^d), relaxing a standard assumption in classical Fourier quasicrystal theory.