On Ruzsa's conjecture on congruence preserving functions

TL;DR

The paper advances Ruzsa's conjecture by proving a new partial case: if a primary pseudo-polynomial $(a_n)$ with $\limsup_{n\to\infty}|a_n|^{1/n}<e$ has generating series $f=\sum_{n\ge0} a_n x^n$ with at most two singular directions at $0$, then $(a_n)$ is a polynomial sequence. The authors adapt Carlson's Pólya–Carlson framework, combining an archimedean bound on Hankel determinants via Pólya’s inequality and transfinite diameter arguments with a non-archimedean divisibility bound obtained from the binomial transform, showing $f$ must be rational. Consequently, $f$ is the expansion of a rational function whose denominator is a power of $(1-x)$, implying $(a_n)$ is polynomial; this also implies any potential counterexample to Ruzsa's conjecture would require at least three singular directions. The work narrows the landscape of counterexamples and connects congruence-preservation, Hankel-determinant criteria, and Carlson-type dichotomies in a concrete growth regime.

Abstract

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.

Theorems & Definitions (13)

• Conjecture A: Ruzsa Ruz71
• Theorem A: Perelli--Zannier PZ84
• Theorem 1.1
• Theorem B: Kronecker's rationality criterion
• Theorem C: Pólya--Carlson dichotomy
• Lemma 2.1
• Theorem D: Pólya's inequality Pol28
• Theorem E: Dubinin Dub85
• proof : Proof of Lemma \ref{['lem: ArchiBound']}
• Lemma 2.2