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PreprintCC BY 4.0arXiv

Weak mixing and non-recurrence along finitely generated multiplicative semigroups of β„•

  • Zhuowei Liu
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Abstract

Let 𝒫 be a finite nonempty set of primes, and let S𝒫 be the multiplicative semigroup that they generate. We prove that there are an invertible weakly mixing measure-preserving transformation T and a set A of positive measure such that A∩ Tβˆ’rA=βˆ… for every r∈ S𝒫. This gives a negative answer to Bergelson's 1996 question concerning recurrence along {2m3n:m,nβ‰₯1}, and applies to every finitely generated multiplicative semigroup of positive integers. More generally, we give a sufficient condition for weakly mixing non-recurrence based on divisibility properties and separation by finitely many rotations of the circle. This criterion also applies to an infinitely generated multiplicative semigroup not contained in any S𝒫 with 𝒫 finite.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026weak,
  title = {{Weak mixing and non-recurrence along finitely generated multiplicative semigroups of $\mathbb N$}},
  author = {Zhuowei Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33710},
  eprint = {2610.33710},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33710},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Weak mixing and non-recurrence along finitely generated multiplicative semigroups of β„•
AU  - Zhuowei Liu
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.33710
UR  - https://arxiv.org/abs/2610.33710
AB  - Let 𝒫 be a finite nonempty set of primes, and let S_𝒫 be the multiplicative semigroup that they generate. We prove that there are an invertible weakly mixing measure-preserving transformation T and a set A of positive measure such that A∩ T^βˆ’rA=βˆ… for every r∈ S_𝒫. This gives a negative answer to Bergelson's 1996 question concerning recurrence along {2^m3^n:m,nβ‰₯1}, and applies to every finitely generated multiplicative semigroup of positive integers. More generally, we give a sufficient condition for weakly mixing non-recurrence based on divisibility properties and separation by finitely many rotations of the circle. This criterion also applies to an infinitely generated multiplicative semigroup not contained in any S_𝒫 with 𝒫 finite.
ER  -