In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to binary division with remainder algorithm which calculates
- Inputs: $n,d \in \mathbb{Z}$ with $\nu(d) - \nu(n) = j > 0$
- Outputs: $q,r \in \mathbb{Z}$ with $\|q\|_\infty < 2^j$ and $r = n + \frac{q}{2^j} d$ and $\| r\|_2 < \| d \|_2$
the point being that you can choose the 2-adic quotient and remainder to have a 2-adic floating point representation requires the "right" number of digits of precision to represent exactly (where "right" means essentially the same thing as the analogous division with remainder with respect to the usual real absolute value), and the lynch pin of the various arguments is ultimately that no nonzero integer satisfies $\|x\|_\infty \|x\|_2 < 1$.
The book introduces this for a specific goal (the 2-adic version of the fast GCD algorithm), and so things get introduced in an ad-hoc fashion rather than in any systematic theory.
So I'm curious, are there any references that develop the theory of how these norms work together? Or maybe have I failed to recognize a familiar idea in disguise?