Refined additive uncertainty principle

TL;DR

The paper addresses the problem of recovering a discrete signal from incomplete frequency information by refining the additive energy uncertainty principle. It introduces a strengthened principle with explicit correction terms that measure how far the supports are from extremal, highly structured sets (e.g., cosets of subgroups), yielding deterministic improvements over the prior additive-energy bound. Two main results are established: a stronger additive energy uncertainty principle with correction terms $C(E,\Sigma)$ and $C(\Sigma,E)$ that vanish at the extremal case, and a correspondingly improved recovery condition for unique reconstruction from partial Fourier data. These results provide sharper sufficient conditions for exact recovery and clarify the role of additive structure in recoverability, with potential extensions to higher-order energies, Gowers norms, and broader settings beyond the current finite abelian group framework.

Abstract

Signal recovery from incomplete or partial frequency information is a fundamental problem in harmonic analysis and applied mathematics, with wide-ranging applications in communications, imaging, and data science. Historically, the classical uncertainty principles, such as those by Donoho and Stark, have provided essential bounds relating the sparsity of a signal and its Fourier transform, ensuring unique recovery under certain support size constraints. Recent advances have incorporated additive combinatorial notions, notably additive energy, to refine these uncertainty principles and capture deeper structural properties of signal supports. Building upon this line of work, we present a strengthened additive energy uncertainty principle for functions $f:\mathbb{Z}_N^d\to\mathbb{C}$, introducing explicit correction terms that measure how far the supports are from highly structured extremal sets like subgroup cosets. We have two main results. Our first theorem introduces a correction term which strictly improves the additive energy uncertainty principle from Aldahleh et al., provided that the classical uncertainty principle is not satisfied with equality. Our second theorem uses the improvement to obtain a better recovery condition. These theorems deliver strictly improved bounds over prior results whenever the product of the support sizes differs from the ambient dimension, offering a more nuanced understanding of the interplay between additive structure and Fourier sparsity. Importantly, we leverage these improvements to establish sharper sufficient conditions for unique and exact recovery of signals from partially observed frequencies, explicitly quantifying the role of additive energy in recoverability.

Theorems & Definitions (13)

• Theorem 1.1: DonohoStark
• Definition 1.2: Additive Energy, 2025additiveenergyuncertaintyprinciple
• Theorem 1.3: Theorem 1.8, 2025additiveenergyuncertaintyprinciple
• Theorem 1.4: Stronger Additive Uncertainty Principle
• Example 1.5
• Theorem 1.6: Additive Recovery Condition
• Theorem 1.7: Recovery via least squares, Theorem 1.21 burstein2025fourierminimizationimputationtime
• Theorem 1.8: Recovery via Logan's method, Theorem 1.23 burstein2025fourierminimizationimputationtime
• Example 2.1
• proof : Proof of Theorem \ref{['thm:new_principle']}