Improvement of effective Erdos-Wintner theorem for Zeckendorf expansions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909881502531584 |
|---|---|
| author | Verwee, Johann |
| author_facet | Verwee, Johann |
| contents | We revisit the effective Erdos-Wintner theorem for Zeckendorf expansions. Drmota and the author obtained a uniform Kolmogorov bound whose error involves $T\sum_{j>L-2h}|f(F_j)|$, which assumes absolute convergence of the linear tail $\sum_j f(F_j)$. We remove this assumption. Grouping the transfer matrices in pairs and working to second order on the logarithm of the product, after extracting the common linear phase along the dominant direction, yields a quadratic tail $T^2\sum_{j>L-2h} f(F_j)^2$, or, in a flexible variant, the split tail $T\sum_{|f(F_j)|>1/T}|f(F_j)| + T^2\sum_{|f(F_j)|\le 1/T} f(F_j)^2$. Either form requires only $\sum f(F_j)^2<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14974 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improvement of effective Erdos-Wintner theorem for Zeckendorf expansions Verwee, Johann Number Theory Combinatorics Probability 11K65 (Primary) 11B39, 60F05, 60E10 (Secondary) We revisit the effective Erdos-Wintner theorem for Zeckendorf expansions. Drmota and the author obtained a uniform Kolmogorov bound whose error involves $T\sum_{j>L-2h}|f(F_j)|$, which assumes absolute convergence of the linear tail $\sum_j f(F_j)$. We remove this assumption. Grouping the transfer matrices in pairs and working to second order on the logarithm of the product, after extracting the common linear phase along the dominant direction, yields a quadratic tail $T^2\sum_{j>L-2h} f(F_j)^2$, or, in a flexible variant, the split tail $T\sum_{|f(F_j)|>1/T}|f(F_j)| + T^2\sum_{|f(F_j)|\le 1/T} f(F_j)^2$. Either form requires only $\sum f(F_j)^2<\infty$. |
| title | Improvement of effective Erdos-Wintner theorem for Zeckendorf expansions |
| topic | Number Theory Combinatorics Probability 11K65 (Primary) 11B39, 60F05, 60E10 (Secondary) |
| url | https://arxiv.org/abs/2509.14974 |