On Ruzsa's conjecture on congruence preserving functions

2025/02/18 by Éric Delaygue, Delaygue, É.
Mathematics · #Mathematical functions and polynomials #Advanced Mathematical Identities #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2502.13068

Abstract

Ruzsa's conjecture asserts that any sequence (a_n)_{n ≥ 0} of integers that preserves congruences, i.e., satisfies a_n+k ≡ a_n \mod k , and has the growth condition \limsup_{n → +∞} |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 = ∑_{n ≥ 0} a_n xⁿ has at most two singular directions at x = 0 , then (a_n)_{n ≥ 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.

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