S-adic version of Minkowski's geometry of numbers and Mahler's   compactness criterion
  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. 
Dmitry Kleinbock, Ronggang Shi, George Tomanov
math.DS