Universal completeness of exponentials
Abstract
We consider generalizations of the classical Fourier uniqueness theorem. First, we construct a family of uniformly discrete sets Λ⊂ ℝ, of uniform density one, such that the exponential system {e2πiλx : λ∈ Λ} is complete in Lp(S) for every 1 ≤ p < ∞ and every measurable set S ⊂ ℝ with |S| < 1. We also show that no set that is asymptotically integer can have this universality property. Additionally, for every v ∈ (0,1), we construct a set of integer frequencies and uniform density v whose exponential system is complete in Lp(S) for every 1 ≤ p < ∞ and every measurable set S ⊂ [0,1] with |S| < v. Finally, we prove that the Sobolev regularity condition α> 1/2 for the existence of uniformly discrete uniqueness sets for spectra with periodic weak gaps, considered by Olevskii and Ulanovskii, is sharp. Our findings admit extensions to higher dimensions and are verified in Lean.
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BibTeX
@misc{bertolini2026universal,
title = {{Universal completeness of exponentials}},
author = {Susanna Bertolini and Enric Florit-Simon and Lukas Liehr and Mitchell A. Taylor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20805},
eprint = {2609.20805},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20805},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universal completeness of exponentials
AU - Susanna Bertolini
AU - Enric Florit-Simon
AU - Lukas Liehr
AU - Mitchell A. Taylor
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20805
UR - https://arxiv.org/abs/2609.20805
AB - We consider generalizations of the classical Fourier uniqueness theorem. First, we construct a family of uniformly discrete sets Λ⊂ ℝ, of uniform density one, such that the exponential system {e^2πiλx : λ∈ Λ} is complete in L^p(S) for every 1 ≤ p < ∞ and every measurable set S ⊂ ℝ with |S| < 1. We also show that no set that is asymptotically integer can have this universality property. Additionally, for every v ∈ (0,1), we construct a set of integer frequencies and uniform density v whose exponential system is complete in L^p(S) for every 1 ≤ p < ∞ and every measurable set S ⊂ [0,1] with |S| < v. Finally, we prove that the Sobolev regularity condition α> 1/2 for the existence of uniformly discrete uniqueness sets for spectra with periodic weak gaps, considered by Olevskii and Ulanovskii, is sharp. Our findings admit extensions to higher dimensions and are verified in Lean.
ER -