Khintchin conjecture and related topics
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908757012774912 |
|---|---|
| author | Fan, Aihua Fan, Shilei Queffélec, Hervé Queffélec, Martine |
| author_facet | Fan, Aihua Fan, Shilei Queffélec, Hervé Queffélec, Martine |
| contents | Motivated by Khintchin's 1923 conjecture, refuted by Marstrand in 1970, we study the Khintchin class of functions associated to a given increasing sequence of integers. When the Khintchin class contains L^p(\mathbb{T}), we call the sequence a L^p-Khintchin sequence. We establish basic properties of Khintchin sequences, provide several constructions, and propose open problems for further research. We also initiate the study of Khintchin sequences of group endomorphisms on compact abelian groups. Under a Fourier-tightness assumption, we show that ergodicity (respectively, weakly mixing or strongly mixing) of a skew product of endomorphisms is equivalent to the corresponding property of the base system, supporting the idea that typical fiber orbits in such skew products should form Khintchin sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06626 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Khintchin conjecture and related topics Fan, Aihua Fan, Shilei Queffélec, Hervé Queffélec, Martine Dynamical Systems 11J71, 28D05 Motivated by Khintchin's 1923 conjecture, refuted by Marstrand in 1970, we study the Khintchin class of functions associated to a given increasing sequence of integers. When the Khintchin class contains L^p(\mathbb{T}), we call the sequence a L^p-Khintchin sequence. We establish basic properties of Khintchin sequences, provide several constructions, and propose open problems for further research. We also initiate the study of Khintchin sequences of group endomorphisms on compact abelian groups. Under a Fourier-tightness assumption, we show that ergodicity (respectively, weakly mixing or strongly mixing) of a skew product of endomorphisms is equivalent to the corresponding property of the base system, supporting the idea that typical fiber orbits in such skew products should form Khintchin sequences. |
| title | Khintchin conjecture and related topics |
| topic | Dynamical Systems 11J71, 28D05 |
| url | https://arxiv.org/abs/2601.06626 |