Limit Theorems for $θ$-expansions and the Failure of the Strong Law
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912833617264640 |
|---|---|
| author | Rusu, Andreas Sebe, Gabriela Ileana Lascu, Dan |
| author_facet | Rusu, Andreas Sebe, Gabriela Ileana Lascu, Dan |
| contents | The paper presents fundamental metrical theorems for a class of continued fraction-like expansions known as $θ$-expansions. We first prove Khinchine's Weak Law of Large Numbers for the sum of digits, followed by the Diamond-Vaaler Strong Law for the sum of digits minus the largest one. Our main result is a general theorem on the failure of the strong law, showing that no regular norming sequence can yield a finite, non-zero almost sure limit. This result extends a classical theorem of Philipp to the $θ$-expansion setting. The proofs leverage the system's explicit invariant measure and a detailed analysis of its mixing properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13296 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Limit Theorems for $θ$-expansions and the Failure of the Strong Law Rusu, Andreas Sebe, Gabriela Ileana Lascu, Dan Number Theory 11K55, 37A05, 60F05, 60F15 The paper presents fundamental metrical theorems for a class of continued fraction-like expansions known as $θ$-expansions. We first prove Khinchine's Weak Law of Large Numbers for the sum of digits, followed by the Diamond-Vaaler Strong Law for the sum of digits minus the largest one. Our main result is a general theorem on the failure of the strong law, showing that no regular norming sequence can yield a finite, non-zero almost sure limit. This result extends a classical theorem of Philipp to the $θ$-expansion setting. The proofs leverage the system's explicit invariant measure and a detailed analysis of its mixing properties. |
| title | Limit Theorems for $θ$-expansions and the Failure of the Strong Law |
| topic | Number Theory 11K55, 37A05, 60F05, 60F15 |
| url | https://arxiv.org/abs/2601.13296 |