论文标题
关于素的多项式，阵尾平均值和一多裂群
On Polynomials in Primes, Ergodic Averages and Monothetic Groups
论文作者
论文摘要
令$ g $表示一个紧凑的一分子群，然后让$$ρ（x）=α_kx^k + \ ldots +α_1x +α_0，$ $ $α_0，\ ldots，α_k$是$ g $的元素，其中一个是$ g $的生成器。令$（p_n）_ {n \ geq 1} $表示有理数的序列。假设$ f \ in l^{p}（g）$ for $ p> 1 $。众所周知，如果$$ a_ {n} f（x）：= {1 \ aver n} \ sum_ {n = 1}^{n} f（x +ρ（p_n））\ qquad（n = 1,2，\ ldots）哈尔措施。我们表明，如果连接$ g $，则限制为$ \ int_ {g} fdλ$。如果$ g $是$ a $ a-adic Integers，这是一个完全断开的组，则限制是用傅立叶乘数来描述的，这些乘数是高斯总和的概括。
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.
