Constructing topological groups through unit equations (Diophantine Problems and Analytic Number Theory)

We treat problems concerning duality properties of topological gouffi $\cdot$ To solve them, we make the additive group of the integers into topological groups.The onstruction depends on afamily of exponential Diophantine equations.1IntroductionWe exhibit an application of exponential Diophantine equations to some prok lems on characters of topological groups.In Section 2, we introduce two duality properties we consider.Section 3is for the explanation of the metrics on the integers due to J. W. Nienuys [4].In Section 4, we find particular metrics answering the questions.The construction is closely tied with afamily of 5-unit equations.As an appendix, we mention the ineffectiveness of the method.Most of the contents of this article overlap those of [5] or [6], which is mainly intended for the audience with atopological badcground.Here we proceed more number-theoretically.