Let $ W $ be a linear subspace of $ \mathbb{R}^n $.
Let $ \Gamma $ be the $ \mathbb{Z} $ span of some set of linearly independent vectors $ v_1, \dots, v_k $ in $ \mathbb{R}^n $.
Is there any efficient way to find a nonzero point in the intersection $ W \cap \Gamma $ (or even to tell if the intersection is nontrival)?
(This $ \Gamma $ is what I'm referring to as a lattice in the question title, although it is technically only a lattice if $ k = n $.)
I'm also happy to assume the $ v_1, \dots, v_k $ are orthogonal if that helps.
My thoughts so far are to view $ W $ as the kernel of some linear transformation $ M $. Then we want some choice integers $ n_i $ not all $ 0 $ such that $$ \sum_{i=1}^k n_i v_i \in W $$ is equivalent to $$ M \Big( \sum_{i=1}^k n_i v_i \Big) =0 $$ $$ \sum_{i=1}^k n_i M v_i =0 $$ which is the same as finding an integer linear dependence relation between the $ Mv_i $. Is there a well known method/efficient algorithm for find an integer dependence relation between a set of vectors in $ \mathbb{R}^n$?
Seems like the first step would be to test if the set $ Mv_1, \dots Mv_k $ is linearly independent. If it is then just give up because there are no dependence relations, let alone integer ones.
If the set is dependent then how does one proceed searching for integer dependence relations?
If it helps this question is inspired by a problem in representation theory of finite groups so everything is really being done over a finite degree extension of $ \mathbb{Q} $ rather than truly being done over $ \mathbb{R} $.