On Polynomials in Primes, Ergodic Averages and Monothetic Groups

Let $G$ denote a compact monothetic group, and let $ρ(x) = α_k x^k + \ldots + α_1 x + α_0,$ where $α_0, \ldots , α_k$ are elements of $G$ one of which is a generator of $G$. Let $(p_n)_{n\geq 1}$ denote the sequence of rational prime numbers. Suppose $f \in L^{p}(G)$ for $p> 1$. It is known that if $A_{N}f(x) := {1 \over N} \sum_{n=1}^{N} f(x + ρ(p_n)) \qquad (N=1,2, \ldots ),$ then the limit $\lim _{n\to \infty} A_Nf(x)$ exists for almost all $x$ with respect Haar measure. We show that if $G$ is connected then the limit is $\int_{G} f dλ$. 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.

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