Subadditive and submultiplicative norms on $\mathbb{Q}$

Question

Say a norm on a ring must satisfy :

• $\|x\| \in [0,\infty)$

• $\|x+y\| \le \|x\|+\|y\|$

• $\|xy\| \le \|x\|\|y\|$

• $\|x\| = 0 \Longleftrightarrow x= 0$

• $\|1\| = 1$

Note we don't require that $\|xy\| = \|x\| \|y\|$ (it would make it an absolute value, which is needed for Ostrowski's theorem).

1. How to find all the norms on $\mathbb{Z}$; there is the usual one, the $\ell$-adic ones, but are there others?

2. How to find all the norms on $\mathbb{Q}$ ?

3. Which ones satisfy $\# \{ x \in \mathbb{Q},\|x\| < r\} < \infty$ ?

The $\rho$-adic absolute values are of the form $|\frac{u}{v}\rho^{-k}|_{\rho,\alpha} = \alpha^{k}$ whenever $u,v\in \mathbb{Z}^*,$ $k \in \mathbb{Z},$ $\rho \nmid u,$ and $\gcd(v,\rho)=1$ for some $\alpha \ge 1$ and $\ell$ a prime number. When unspecified we set $\alpha = \rho$. For $\rho$ not a prime, it is a norm but not an absolute value.

Let $|\cdot|_\infty$ be the usual absolute value on $\mathbb{R}$.

The "adelic norm" $\|x\|_{\mathbb{A}} = \max(|x|_\infty, \sup_{p \text{ prime}} |x|_{p})$ satisfies 3.

Maybe it would help to think this way : what is the set of functions $f$ such that $x \mapsto f(|x|_\infty,|x|_{2},|x|_3, |x|_5,|x|_7,\ldots)$ is a norm ?