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The Shubin--Vakilian--Wolff Uncertainty Principle at Half Density

  • Ming Wang
  • Yunlei Wang
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Abstract

Shubin, Vakilian, and Wolff proved a Fourier uncertainty principle for sets of sufficiently small local density at the reciprocal scale and asked whether every density below one is admissible. We answer this question negatively in dimension one by constructing sequences of pairs of 1/2-density thin sets and unit vectors whose total position and Fourier mass outside these sets tends to zero. This obstruction persists for every scaled reciprocal profile ρκ(x)=min{1,κ/|x|}, κ>0. On the other hand, for 0<κ≤1, the uncertainty estimate holds whenever the density ε<1/(2(1+32κ)). Thus the critical density tends to 1/2 as κ↓ 0. The obstruction uses odd Gaussian packets to transfer norm bounds from a free-group model. The positive estimate uses a Fourier-complementary anti-Wick operator and quadratic straightening of the reciprocal geometry.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026shubinvakilianwolff,
  title = {{The Shubin--Vakilian--Wolff Uncertainty Principle at Half Density}},
  author = {Ming Wang and Yunlei Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.13958},
  eprint = {2609.13958},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13958},
}
RIS
TY  - UNPB
TI  - The Shubin--Vakilian--Wolff Uncertainty Principle at Half Density
AU  - Ming Wang
AU  - Yunlei Wang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.13958
UR  - https://arxiv.org/abs/2609.13958
AB  - Shubin, Vakilian, and Wolff proved a Fourier uncertainty principle for sets of sufficiently small local density at the reciprocal scale and asked whether every density below one is admissible. We answer this question negatively in dimension one by constructing sequences of pairs of 1/2-density thin sets and unit vectors whose total position and Fourier mass outside these sets tends to zero. This obstruction persists for every scaled reciprocal profile ρ_κ(x)=min{1,κ/|x|}, κ>0. On the other hand, for 0<κ≤1, the uncertainty estimate holds whenever the density ε<1/(2(1+32κ)). Thus the critical density tends to 1/2 as κ↓ 0. The obstruction uses odd Gaussian packets to transfer norm bounds from a free-group model. The positive estimate uses a Fourier-complementary anti-Wick operator and quadratic straightening of the reciprocal geometry.
ER  -