  We study the problem of best approximations of a vector $\alpha\in{\mathbb
R}^n$ by rational vectors of a lattice $\Lambda\subset {\mathbb R}^n$ whose
common denominator is bounded. To this end we introduce successive minima for a
periodic lattice structure and extend some classical results from geometry of
numbers to this structure. This leads to bounds for the best approximation
problem which generalize and improve former results.
