On periodic \(\text{mod }p\) sequences and \(G\)-functions. (On a conjecture of Ruzsa) (Q1817362)

Let the function \(f:\mathbb{N} \to\mathbb{Z}\) satisfy (i) \(f(n+b) \equiv f(n) \bmod b\) for all natural \(b,n\); (ii) \(f(n)= O(e^{\alpha n})\) where \(\alpha<1\). Then I. Z. Ruzsa conjectured that \(f\) is a polynomial. He provided this with \(e-1\) instead of \(e\) in (ii). In this paper the author proves it with \(\exp (0,75) \approx 2,1\) and (i) for all large primes \(p\) instead of \(b\). He uses the arithmetic theory of \(G\)-functions.