Abstract

Let \(p\) and \(q\) be distinct primes, and consider the expression \(S_{p,q}\left(\mathfrak{z}\right)\) defined by the formal series \(\sum_{n=0}^{\infty}q^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}/p^{n}\), where \(\mathfrak{z}\) is a \(2\)-adic integer variable, \(\left[\mathfrak{z}\right]_{2^{n}}\) is the integer in \(\left\{ 0,\ldots,2^{n}-1\right\} \) congruent to \(\mathfrak{z}\) mod \(2^{n}\), and where, for any integer \(m\geq0\), \(\#_{1}\left(m\right)\) is the number of \(1\)s in the binary expansion of \(m\). When \(\mathfrak{z}\in\left\{ 0,1,2,\ldots\right\} \), \(S_{p,q}\left(\mathfrak{z}\right)\) reduces to a geometric series with \(1/p\) as its common ratio. This series converges in the topology of \(\mathbb{R}\), and its sum is a rational number which is also a \(q\)-adic integer. For \(\mathfrak{z}\in\mathbb{Z}_{2}\backslash\left\{ 0,1,2,\ldots\right\} \), \(\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)\rightarrow\infty\) as \(n\rightarrow\infty\), and so \(S_{p,q}\left(\mathfrak{z}\right)\) converges in the \(q\)-adic topology to a \(q\)-adic integer. In this way, we can define a “\(\left(2,q\right)\)-adic function” \(X_{p,q}:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{q}\) by the sum of \(S_{p,q}\left(\mathfrak{z}\right)\) in \(\mathbb{R}\) for \(\mathfrak{z}\in\mathbb{Z}_{2}\backslash\left\{ 0,1,2,\ldots\right\} \) and as the sum of \(S_{p,q}\left(\mathfrak{z}\right)\) in \(\mathbb{Z}_{q}\) for all other \(\mathfrak{z}\). Thus, while \(X_{p,q}\left(\mathfrak{z}\right)\) is well-defined as a \(q\)-adic integer for all \(\mathfrak{z}\in\mathbb{Z}_{2}\), the topology of convergence used to sum the series representation of \(X_{p,q}\left(\mathfrak{z}\right)\) to compute its value at any given \(\mathfrak{z}\) depends on the value of \(\mathfrak{z}\). This represents an entirely new type of point-wise convergence, one where the topology in which the limit of a sequence of functions \(\left\{ f_{n}\right\} _{n\geq1}\) is evaluated depends on the point at which the sequence is evaluated. In a manner comparable to the adèle ring of a number field, functions defined by \(\mathcal{F}\)-series require considering different metric completions of an underlying field in order to be properly understood. This paper catalogues a variety of examples of this new, unstudied convergence phenomenon, and presents the concept of a “frame”, a rigorous formalism for defining and studying \(\mathcal{F}\)-series and their peculiar modes of convergence.

中文翻译：

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收敛拓扑逐点变化的无穷级数

摘要

设\(p\)和\(q\)是不同的素数，并考虑由形式级数\(\sum_定义的表达式\(S_{p,q}\left(\mathfrak{z}\right)\) {n=0}^{\infty}q^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}/p^{n }\)，其中\(\mathfrak{z}\)是一个\(2\)进整数变量，\(\left[\mathfrak{z}\right]_{2^{n}}\)是\(\left\{ 0,\ldots,2^{n}-1\right\} \)中的整数与\(\mathfrak{z}\) mod \(2^{n}\)一致，并且其中，对于任何整数\(m\geq0\)，\(\#_{1}\left(m\right)\)是\(1\)的个数s 是\(m\)的二进制展开式。当\(\mathfrak{z}\in\left\{ 0,1,2,\ldots\right\} \)时，\(S_{p,q}\left(\mathfrak{z}\right)\)简化为以\(1/p\)为公比的几何级数。该级数收敛于\(\mathbb{R}\)拓扑，其和是一个有理数，也是一个\(q\)进整数。对于\(\mathfrak{z}\in\mathbb{Z}_{2}\backslash\left\{ 0,1,2,\ldots\right\} \) ， \(\#_{1} \ left (\left[\mathfrak{z}\right]_{2^{n}}\right)\rightarrow\infty\)为\(n\rightarrow\infty\)，因此\(S_{p,q} \left(\mathfrak{z}\right)\)收敛于\(q\)-adic 拓扑到\(q\) -adic 整数。这样，我们就可以定义一个“ \(\left(2,q\right)\) -adic 函数” \(X_{p,q}:\mathbb{Z}_{2}\rightarrow\mathbb{Z }_{q}\)除以\(\mathfrak{z}\) 中\(\mathbb{R}\)中的\(S_{p,q}\left( \mathfrak{z}\right)\)之和\in\mathbb{Z}_{2}\backslash\left\{ 0,1,2,\ldots\right\} \)和\(S_{p,q}\left(\mathfrak{ z}\right)\)中的\(\mathbb{Z}_{q}\)对于所有其他\(\mathfrak{z}\)。因此，虽然\(X_{p,q}\left(\mathfrak{z}\right)\)被明确定义为所有\(\mathfrak{z}\in\)的\(q\) -adic 整数mathbb{Z}_{2}\)，用于对 \(X_{p,q}\left(\mathfrak{z}\right)\) 的级数表示求和以计算其在任何给定\(\mathfrak{z}\)处的值的收敛拓扑取决于关于\(\mathfrak{z}\)的值。这代表了一种全新类型的逐点收敛，其中评估函数序列\(\left\{ f_{n}\right\} _{n\geq1}\) 极限的拓扑取决于在评估序列的点上。类似于数域的 adèle 环，由\(\mathcal{F}\)定义的函数-系列需要考虑基础字段的不同度量完成才能正确理解。本文列出了这种新的、未经研究的收敛现象的各种例子，并提出了“框架”的概念，这是一种用于定义和研究 \(\mathcal{F}\)级数及其特殊收敛模式的严格形式主义。