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, $where $\alpha_0, \ldots , \alpha_k$ 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 + \rho (p_n)) \qquad (N=1,2, \ldots ),$ $t h e n t h e l i m i t$ \lim _{n\to \infty} A_Nf(x) $e x i s t s f o r a l m o s t a l l$ x $w i t h r e s p e c t H a a r m e a s u r e . W e s h o w t h a t i f$ G$ is connected then the limit is $\int_{G} f d λ$ . 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.

Let $G$ denote a compact monothetic group, and let $$\rho (x) = \alpha_k x^k

\ldots + \alpha_1 x + \alpha_0, $where $\alpha_0, \ldots , \alpha_k$ 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 + \rho (p_n)) \qquad (N=1,2, \ldots ),$ $t h e n t h e l i m i t$ \lim _{n\to \infty} A_Nf(x) $e x i s t s f o r a l m o s t a l l$ x $w i t h r e s p e c t H a a r m e a s u r e . W e s h o w t h a t i f$ G$ is connected then the limit is $\int_{G} f d λ$ . 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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