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PreprintarXiv licencearXiv

On the identification of certain classes of gap-rich sets

  • Jake Fillman
  • Yi-Sheng Lim
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Abstract

The notion of gap-richness was recently introduced to quantify the ability to open spectral gaps for periodic word models by arbitrarily small perturbations and is useful in the construction of almost-periodic operators having exotic spectral properties, such as Cantor spectrum of zero Hausdorff dimension. Given a subset of Euclidean space that is locally dense in the Zariski topology, we show that any image of such a set under a nonconstant polynomial is gap-rich. This generalizes the scope of previous work in several ways, including allowing local correlations into potential blocks and allowing for totally disconnected sets of realizations.

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Cite

BibTeXRIS
BibTeX
@misc{fillman2026identification,
  title = {{On the identification of certain classes of gap-rich sets}},
  author = {Jake Fillman and Yi-Sheng Lim},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21029},
  eprint = {2609.21029},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21029},
}
RIS
TY  - UNPB
TI  - On the identification of certain classes of gap-rich sets
AU  - Jake Fillman
AU  - Yi-Sheng Lim
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.21029
UR  - https://arxiv.org/abs/2609.21029
AB  - The notion of gap-richness was recently introduced to quantify the ability to open spectral gaps for periodic word models by arbitrarily small perturbations and is useful in the construction of almost-periodic operators having exotic spectral properties, such as Cantor spectrum of zero Hausdorff dimension. Given a subset of Euclidean space that is locally dense in the Zariski topology, we show that any image of such a set under a nonconstant polynomial is gap-rich. This generalizes the scope of previous work in several ways, including allowing local correlations into potential blocks and allowing for totally disconnected sets of realizations.
ER  -