On polynomials in primes, ergodic averages and monothetic groups

Let G denote a compact monothetic group, and let \(\rho (x) = \alpha _k x^k + \ldots + \alpha _1 x + \alpha _0\) $\rho (x)={\alpha }_k{x}^k+\dots +{\alpha }_{1}x+{\alpha }_0$ , where \(\alpha _0, \ldots , \alpha _k\) ${\alpha }_0,\dots ,{\alpha }_k$ are elements of G one of which is a generator of G. Let \((p_n)_{n\ge 1}\) ${(p_n)}_{n\ge 1}$ denote the sequence of rational prime numbers. Suppose \(f \in L^{p}(G)\) $f\in L^p(G)$ for \(p> 1\) $p>1$ . It is known that if \(\begin{aligned} A_{N}f(x):= {1 \over N} \sum _{n=1}^{N} f(x + \rho (p_n)) \quad (N=1,2, \ldots ), \end{aligned}\) $$\begin{array}{c}\hfill A_{N}f(x):=\frac{1}{N}\sum _{n=1}^{N}f(x+\rho (p_n))(N=1,2,\dots ),\end{array}$$ then the limit \(\lim _{n\rightarrow \infty } A_Nf(x)\) ${lim}_{n\to \infty }{A}_{N}f(x)$ exists for almost all x with respect Haar measure. We show that if G is connected then the limit is \(\int _{G} f d\lambda \) ${\int }_{G}fd\lambda$ . 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.