Infinite Series Whose Topology of Convergence Varies From Point to Point

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.