On Ruzsa's conjecture on congruence preserving functions

TL;DR

This paper proves that congruence-preserving integer sequences with generating series having at most two singular directions are polynomial, advancing understanding of Ruzsa's conjecture by linking singular directions to polynomiality.

Contribution

It establishes a new partial result confirming Ruzsa's conjecture under the condition of limited singular directions of the generating series.

Findings

Sequences with at most two singular directions are polynomial.

Counterexamples to Ruzsa's conjecture require at least three singular directions.

The method combines Carlson’s approach with Hankel determinant analysis.

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 P\'olya-Carlson dichotomy, combined with a refined analysis of Hankel determinants. Specifically, we derive an upper bound on these determinants using P\'olya'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.