Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to contain at least one point of the lattice (possibly on the boundary)?
The things I found on the internet all concern the case, in which it is already known, that at least one point of the lattice is contained in the region. Any suggestion would be welcome.