Refined additive uncertainty principle

Ivan Bortnovskyi, June Duvivier, Alex Iosevich, Josh Iosevich, Say-Yeon Kwon, Meiling Laurence, Michael Lucas, Tiancheng Pan, Eyvindur Palsson, Jennifer Smucker and Iana Vranesko

Vol. 345 (2026), No. 1, 1–18

DOI: 10.2140/pjm.2026.345.1

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 (SIAM J. Appl. Math. 49:3 (1989), 906–931) — have provided essential bounds relating the sparsity of a signal and its Fourier transform, ensuring unique recovery under certain support size constraints.

Recent advances incorporate 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 $f:{\mathbb{Z}}_N^d\to ℂ$, introducing explicit correction terms that measure how far the supports are from highly structured extremal sets like subgroup cosets.

We have two main results. The first introduces a correction term which strictly improves the additive energy uncertainty principle of Aldahleh, A. Iosevich, J. Iosevich, Jaimangal, Mayeli, and Pack (arXiv:2504.14702), provided that the classical uncertainty principle is not satisfied with equality. The second 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.

These results advance the theory of discrete signal recovery by providing stronger, more precise guarantees that bridge harmonic analysis and additive combinatorics, and open new pathways for analyzing sparsity and structure in finite discrete settings.

Keywords

signal recovery, additive energy, uncertainty principle

Mathematical Subject Classification

Primary: 42B10, 94A12

Milestones

Received: 9 November 2025

Revised: 10 July 2026

Accepted: 15 July 2026

Published: 25 September 2026