Best Diophantine Approximations and Multidimensional Three Distance Theorem

In 1996 N. Chevallier proved a beautiful lemma which connects Diophantine approximation and multidimensional generalizations of the famous Three Distance Theorem. Using this lemma we show how known results about multidimensional three distance theorem can be deduced from certain known results dealing with the best Diophantine approximations. Also we obtain some new results about liminf version of the problem. Beside this, we discuss the inverse problem: how results about multidimensional three distance theorem can be applied to study best Diophantine approximations.