On Ruzsa's conjecture on congruence preserving functions

Ruzsa's conjecture asserts that any sequence of integers that preserves congruences, that is, satisfies , and has the growth condition , 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 has at most two singular directions at , then 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 . This confirms that counterexamples to Ruzsa's conjecture, if they exist, must exhibit at least three singular directions.