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PreprintCC BY-NC-ND 4.0arXiv

On the measure of definable sets in profinite groups

  • Martina Conte
  • Matteo Vannacci
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Abstract

We study the (Haar) measure of verbal sets in profinite groups, and especially in free profinite groups. Relating the measure of a verbal set with the width of the corresponding word, we find verbal sets of positive measure different from one. We prove a sufficient condition for the measure of a verbal set to be zero in a free profinite group of finite rank, from which we deduce that certain existentially definable sets in free profinite groups of finite rank can only have measure zero or one. Finally, we compute the measure of definable sets in torsion-free finitely generated abelian groups.

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Cite

BibTeXRIS
BibTeX
@misc{conte2026measure,
  title = {{On the measure of definable sets in profinite groups}},
  author = {Martina Conte and Matteo Vannacci},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25448},
  eprint = {2609.25448},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25448},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - On the measure of definable sets in profinite groups
AU  - Martina Conte
AU  - Matteo Vannacci
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.25448
UR  - https://arxiv.org/abs/2609.25448
AB  - We study the (Haar) measure of verbal sets in profinite groups, and especially in free profinite groups. Relating the measure of a verbal set with the width of the corresponding word, we find verbal sets of positive measure different from one. We prove a sufficient condition for the measure of a verbal set to be zero in a free profinite group of finite rank, from which we deduce that certain existentially definable sets in free profinite groups of finite rank can only have measure zero or one. Finally, we compute the measure of definable sets in torsion-free finitely generated abelian groups.
ER  -