Infinite Series Whose Topology of Convergence Varies From Point to Point

TL;DR

This paper introduces $\\mathcal{F}$-series, a new class of functions over $p$-adic integers exhibiting a novel point-wise convergence topology dependent on the evaluation point, requiring multiple metric completions for analysis.

Contribution

It documents a previously unknown form of point-wise convergence for $\mathcal{F}$-series, expanding understanding of convergence behaviors in $p$-adic analysis.

Findings

$\\mathcal{F}$-series exhibit point-dependent convergence topology

Analysis involves multiple metric completions similar to adele rings

Provides examples illustrating the new convergence phenomenon

Abstract

This paper catalogues a variety of examples concerning a type of function of a $p$-adic integer variable defined by a formal series expression we have dubbed "$\mathcal{F}$-series". These series exhibit a new, previously undocumented form of point-wise convergence, one where the topology in which the limit of a sequence of functions $\left\{ f_{n}\right\}_{n\geq1}$ converges depends on the point at which the sequence is evaluated. In a manner comparable to the adele 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.