Here $\mathbb{T}^n := (\mathbb{R} / \mathbb{Z})^n$ is the topological group of the n-dimensional torus and $k \in \mathbb{Z}^n$ is a non-null vector, I'm working about the subgroup

$S = \{x \in \mathbb{T}^n : k \cdot x = 0_{\mathbb{T}^n}\}$

where $\cdot$ is the scalar product. I think that $S$ is isomorphic, as topological group, to $\mathbb{T}^{n-1}$, but I could not prove it.

About the case $n=2$, setting $k = (k_1, k_2)$, $k_2 \neq 0$ I found as a possible candidate isomorphism

$\mathbb{T} \to \mathbb{T}^2 : x \mapsto (k_2 x, \lfloor k_2 x \rfloor / k_2 - k_1 x)$

where $\lfloor \;\rfloor$ is the floor function, however this seems too complicated in the general case, I hope you have useful tips, thanks!