S-adic version of Minkowski's geometry of numbers and Mahler's compactness criterion

TL;DR

This paper provides a detailed proof of known results in the S-adic geometry of numbers, serving as a useful reference for researchers working with these concepts.

Contribution

It offers a comprehensive and accessible proof of S-adic geometry of numbers results, which were previously well-known but not explicitly documented.

Findings

Detailed proof of S-adic Minkowski's geometry of numbers

Clarification of Mahler's compactness criterion in the S-adic setting

Serves as a reference for researchers in the field

Abstract

In this note we give a detailed proof of certain results on geometry of numbers in the $S$-adic case. These results are well-known to experts, so the aim here is to provide a convenient reference for the people who need to use them.