I'm wondering about the time complexity for the closest vector problem and the shortest vector problem with a twist:
This is the statement for the SVP I wish to consider:
$$\lambda(H) = \min_{v \in H- {\mathbf\lbrace{1}\rbrace}} ||v||_{L^2}$$
$H = w_1^{\mathbb Z} \odot w_2^{\mathbb Z},$ where $w_1=(a,1)$ and $w_2=(1,b)$ for some $a,b>0; a,b\neq 1,$ and $(x_1,y_1)\odot(x_2,y_2)=(x_1 x_2, y_1 y_2)$ and $w^{\mathbb Z} = \{ w^n \mid n \in \mathbb{Z} \}.$ And $L^2$ is the Euclidean norm.
The vector, crucially has to be non-zero or it's trivial. Here is some info on the closest vector problem.
I think this can be described as measuring a multiplicative lattice using the norm from the additive lattice.
How does this compare? Is this problem any harder or easier than the standard shortest/closest vector problems?