ON PERIODIC MOD-P SEQUENCES AND G-FUNCTIONS (ON A CONJECTURE OF RUZSA)

I.Z.Ruzsa conjectured that if a function f : N --> Z satisfies \f(n)\ much less than e(alpha n), where alpha < 1, and the congruence f(n + b ) = f(n) (mod b) for all n, b is an element of N, then f is a polynomi al. He actually proved the conclusion with e-1 in place of e, a result that was improved in 1984 by A.Perelli and U.Zannier, who used a diff erent method to replace e-l by a larger number. Combining that method with certain theorems from the arithmetic theory of G-functions, we fu rther relax the necessary upper bound on f.