I have a set of nonzero vectors in ${\mathbb{R}}^{n}$, where for each vector, its nonzero elements have the same magnitude. For example, when $n=4$, $(1,0,1,0)$ and $(-3,3,0,3)$ are in this set, while $(1,2,3,2)$ and $(0,0,0,0)$ are not.

The formal notation I came up for it is

$\{x\in {\mathbb{R}}^{n}\mid x\ne 0\text{and}{x}_{i}\in \{0,k,-k\}\text{for some}k\in \mathbb{R}\},$

but I'm not sure if the "for some k" part is correct. Should it be "for all k" instead?

The formal notation I came up for it is

$\{x\in {\mathbb{R}}^{n}\mid x\ne 0\text{and}{x}_{i}\in \{0,k,-k\}\text{for some}k\in \mathbb{R}\},$

but I'm not sure if the "for some k" part is correct. Should it be "for all k" instead?