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PreprintarXiv licencearXiv

The local dynamical structure of Δ* sets via a new Furstenberg family algebra

  • Angelina Blahodatna
  • Lauren Detmold
  • Daniel Glasscock
  • Anh N. Le
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Abstract

In this paper, we strengthen the connection between the combinatorics of difference sets and the dynamics of group rotations. Our main result shows that sets which have non-empty intersection with all difference subsets of a commutative semigroup possess local Bohr structure. This generalizes results of Bergelson, Furstenberg, and Weiss and Host and Kra from the integers to arbitrary commutative semigroups. We accomplish this by A) utilizing a recent result showing that the regionally proximal relation is an equivalence relation for minimal actions of commutative semigroups and by B) describing a new, DeMorgan-type algebra on Furstenberg families that allows for efficient manipulation and computation. We formally verify all of the results in this paper in Lean. The main results are verified in a Palomar submission, and we link to a Github repository containing code for the complete verification.

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Cite

BibTeXRIS
BibTeX
@misc{blahodatna2026local,
  title = {{The local dynamical structure of $Δ^*$ sets via a new Furstenberg family algebra}},
  author = {Angelina Blahodatna and Lauren Detmold and Daniel Glasscock and Anh N. Le},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10416},
  eprint = {2610.10416},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10416},
}
RIS
TY  - UNPB
TI  - The local dynamical structure of Δ^* sets via a new Furstenberg family algebra
AU  - Angelina Blahodatna
AU  - Lauren Detmold
AU  - Daniel Glasscock
AU  - Anh N. Le
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10416
UR  - https://arxiv.org/abs/2610.10416
AB  - In this paper, we strengthen the connection between the combinatorics of difference sets and the dynamics of group rotations. Our main result shows that sets which have non-empty intersection with all difference subsets of a commutative semigroup possess local Bohr structure. This generalizes results of Bergelson, Furstenberg, and Weiss and Host and Kra from the integers to arbitrary commutative semigroups. We accomplish this by A) utilizing a recent result showing that the regionally proximal relation is an equivalence relation for minimal actions of commutative semigroups and by B) describing a new, DeMorgan-type algebra on Furstenberg families that allows for efficient manipulation and computation. We formally verify all of the results in this paper in Lean. The main results are verified in a Palomar submission, and we link to a Github repository containing code for the complete verification.
ER  -