On polynomials in primes, ergodic averages and monothetic groups

Let G denote a compact monothetic group, and let rho(x)=alpha kx(k)+...+alpha 1 x+alpha 0, where alpha(0),...,alpha(k) are elements of Gone of which is a generator of G. Let(p(n)) n >= 1denote the sequence of rational prime numbers. Suppose f is an element of L (p) (G)for p >1. It is known that if A (N) f(x):=1 /N (N) & sum; (n=1)f(x+rho(p(n))) (N=1,2,...), then the limit lim(n ->infinity)A(N) f(x)exists for almost all x with respect Haar measure. We show that if G is connected then the limit is integral(G)f d lambda. In the case where G is the a-adic integers, which is a totally disconnected group, the limit is described in terms of Fourier multipliers which are generalizations of Gauss sums