# On the existence of certain seminormed groups

Let us define a seminormed group as a triple $\mathcal G = (\mathbb G, \|\cdot\|, \phi)$ consisting of a group $\mathbb G$, a function $\|\cdot\|: \mathbb G \to \mathbb R_0^+$, here termed a (group) seminorm (on $\mathbb G$), and a homomorphism $\phi$ of monoids with zero from $(\mathbb Q, \cdot)$ to $(\mathbb R_0^+, \cdot)$, here called a "monoidal absolute value", such that $\|x+y\| \le \|x\| + \|y\|$ for $x,y \in \mathbb G$ and $\|nx\| = \phi(n) \cdot \|x\|$ for $n \in \mathbb Z$ and $x \in \mathbb G$. Note that $\phi(-n) = \phi(n)$ for all $n \in \mathbb Z$, while $\phi(n) \le |n|$ unless $\|\cdot\|$ is identically zero.

Now, while Ostrowski's theorem tells us that there are, in some sense, "few" absolute values on the rational field, and proves that, up to a certain equivalence, just one of these has dense range, things are quite, quite different for monoidal absolute values (after all, we are almost completely disregarding the additive structure of $\mathbb Q$).

In particular, if $\mathcal W = (w_p)_{p \in \mathbb P}$ is a set of real "weights" indexed by the primes $\mathbb P$, it is clear that the function $\phi_\mathcal{W}: \mathbb Q \to \mathbb R_0^+$ defined by $\phi_\mathcal{W}(0) := 0$ and $\phi_\mathcal{W}(q) := \prod_p 2^{w_p e_p(q)}$ for $q \in \mathbb Q \setminus \{0\}$, where $e_p$ is the usual $p$-adic valuation on $\mathbb Q$, is a monoidal absolute value (yes, the base 2 in this definition has nothing special, and can be replaced by any other positive real number).

We say that $\mathcal W$ is a dense system of weights (shortly, DSW) if the function $\mathbb Q \setminus \{0\} \to \mathbb R: q \mapsto \sum_p w_p e_p(q)$ has dense range, in which case it is trivial to check that $\phi_\mathcal{W}(\mathbb Q)$ is dense. Example: $\mathcal W$ is a DSW if it is not eventually constant and $w_p \to 0$ as $p \to \infty$. Note that $\phi_\mathcal{W}$ includes the trivial absolute value and the $p$-adic absolute values as special cases, but neither of these has dense range. So here is my question:

Question. Does there exist any seminormed group of the form $\mathcal G = (\mathbb G, \|\cdot\|, \phi_\mathcal{W})$ for which $\mathcal W$ is a DSW?

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