# Question about the definition of the CVP as an approximation variant

I have a question about the definition of the (Closest Vector Problem) CVP. In the literature you can find for example this definition of the approximation variant of CVP:

$$CVP_\gamma$$, Search: Given a basis $$B \in \mathbb{Z}^{m \times n}$$ and a point $$t \in \mathbb{Z}^m$$, find a point $$x \in \mathcal{L}(B)$$ such that $$\forall y \in \mathcal{L}(B)$$,$$||x-t|| \leq \gamma|| y-t ||$$

Question:

My question is now and unfortunately this is a bit misleading, since some authors define $$t$$ as an element of $$\mathbb{R}^m$$. What are the arguments that speak for using $$t \in \mathbb{Z}^m$$ in the definition for the CSV instead of $$t \in \mathbb{R}^m$$?

In the hope that I have not formulated my question too unclearly, I would be interested in what you think of it?

• This is perhaps a pedantic point, but one can efficiently represent $t\in\mathbb{Z}^m$ with $\approx m\log_2 \lVert t\rVert_\infty$ bits, while it is not clear you can represent an arbitrary point $t\in\mathbb{R}^m$. Of course, one can always simply choose a good-enough fixed-point approximation of $t\approx \tilde{t}\in\frac{1}{2^k}\mathbb{Z}^m$, hence why this mostly pedantic.
– Mark
Oct 24, 2022 at 5:10