On Polynomials in Primes, Ergodic Averages and Monothetic Groups

Abstract

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

2nd Domain: ergodic theory