On Ruzsa's conjecture on congruence preserving functions
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_ | 1866915850777264128 |
|---|---|
| author | Delaygue, É. |
| author_facet | Delaygue, É. |
| contents | Ruzsa's conjecture asserts that any sequence $(a_n)_{n \geq 0}$ of integers that preserves congruences, $\textit{i.e.}$, satisfies $ a_{n+k} \equiv a_n \mod k $, and has the growth condition $\limsup_{n \to +\infty} |a_n|^{1/n} < e$, must be a polynomial sequence. While previous results by Hall, Ruzsa, Perelli, and Zannier have confirmed this conjecture under stricter growth bounds, the general case remains open. In this paper, we establish a new partial result by proving that if in addition the generating series $ f = \sum_{n \geq 0} a_n x^n $ has at most two singular directions at $ x = 0 $, then $(a_n)_{n \geq 0}$ is necessarily a polynomial sequence. Our approach is based on an adaptation of Carlson's method, originally developed for the Pólya-Carlson dichotomy, combined with a refined analysis of Hankel determinants. Specifically, we derive an upper bound on these determinants using Pólya's inequality and a transfinite diameter argument of Dubinin, while a non-Archimedean divisibility condition on Hankel determinants yields a lower bound, ultimately leading to the rationality of $ f $. This confirms that counterexamples to Ruzsa's conjecture, if they exist, must exhibit at least three singular directions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13068 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Ruzsa's conjecture on congruence preserving functions Delaygue, É. Number Theory 11B50, 11C08 (Primary) 30B40, 11C20 (Secondary) Ruzsa's conjecture asserts that any sequence $(a_n)_{n \geq 0}$ of integers that preserves congruences, $\textit{i.e.}$, satisfies $ a_{n+k} \equiv a_n \mod k $, and has the growth condition $\limsup_{n \to +\infty} |a_n|^{1/n} < e$, must be a polynomial sequence. While previous results by Hall, Ruzsa, Perelli, and Zannier have confirmed this conjecture under stricter growth bounds, the general case remains open. In this paper, we establish a new partial result by proving that if in addition the generating series $ f = \sum_{n \geq 0} a_n x^n $ has at most two singular directions at $ x = 0 $, then $(a_n)_{n \geq 0}$ is necessarily a polynomial sequence. Our approach is based on an adaptation of Carlson's method, originally developed for the Pólya-Carlson dichotomy, combined with a refined analysis of Hankel determinants. Specifically, we derive an upper bound on these determinants using Pólya's inequality and a transfinite diameter argument of Dubinin, while a non-Archimedean divisibility condition on Hankel determinants yields a lower bound, ultimately leading to the rationality of $ f $. This confirms that counterexamples to Ruzsa's conjecture, if they exist, must exhibit at least three singular directions. |
| title | On Ruzsa's conjecture on congruence preserving functions |
| topic | Number Theory 11B50, 11C08 (Primary) 30B40, 11C20 (Secondary) |
| url | https://arxiv.org/abs/2502.13068 |