Almost-everywhere convergence and polynomials

Denote by Gamma the set of pointwise good sequences: sequences of real numbers (a(k)) such that for any measure-preserving flow (U-t)(t is an element of R) on a probability space and for any f is an element of L-infinity, the averages 1/n Sigma(n)(k=1) f (Ua(k)x) converge almost everywhere. We prove the following two results. 1. If f: (0,infinity) -> R is continuous and if (f(ku+ v))(k >= 1)is an element of Gamma for all u, v > 0, then f is a polynomial on some subinterval J subset of (0,infinity) of positive length. 2. If f: (0,infinity) -> R is real analytic and if (f(ku))(k >= 1)is an element of Gamma for all u > 0, then f is a polynomial on the whole domain [0,infinity). These results can be viewed as converses of Bourgain's polynomial ergodic theorem which claims that every polynomial sequence lies in Gamma.