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Separating the Polar of Sets of Uniqueness from the Polars of All H(d)-Sets

  • Václav Vlasák
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Abstract

We construct a perfect set of uniqueness on the circle carrying a continuous probability measure which annihilates every H(d)-set, simultaneously for all finite dimensions d. This gives a strict inclusion between the polar of the compact sets of uniqueness and the polar of the union of the classes H(d), answering a question raised in earlier work on these polars. The construction uses random binary increments of size comparable to the reciprocal of a product of square roots. A Gaussian sampling estimate gives a uniform Fourier-tail bound for all pseudomeasures supported by the resulting set, and also bounded atomic approximation and spectral synthesis. A separate energy estimate, stable under deletion of a small number of increments, yields full support for every limiting quasi-independent image of every positive measure absolutely continuous with respect to the constructed measure.

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Cite

BibTeXRIS
BibTeX
@misc{vlasak2026separating,
  title = {{Separating the Polar of Sets of Uniqueness from the Polars of All $H^(d)$-Sets}},
  author = {Václav Vlasák},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02241},
  eprint = {2611.02241},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02241},
}
RIS
TY  - UNPB
TI  - Separating the Polar of Sets of Uniqueness from the Polars of All H^(d)-Sets
AU  - Václav Vlasák
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02241
UR  - https://arxiv.org/abs/2611.02241
AB  - We construct a perfect set of uniqueness on the circle carrying a continuous probability measure which annihilates every H^(d)-set, simultaneously for all finite dimensions d. This gives a strict inclusion between the polar of the compact sets of uniqueness and the polar of the union of the classes H^(d), answering a question raised in earlier work on these polars. The construction uses random binary increments of size comparable to the reciprocal of a product of square roots. A Gaussian sampling estimate gives a uniform Fourier-tail bound for all pseudomeasures supported by the resulting set, and also bounded atomic approximation and spectral synthesis. A separate energy estimate, stable under deletion of a small number of increments, yields full support for every limiting quasi-independent image of every positive measure absolutely continuous with respect to the constructed measure.
ER  -