  In this paper, we study multi-rotation orbits on the unit circle. We obtain a
natural generalization of a classical result which says that orbits of
irrational rotations on the unit circle are dense. It is possible to show that
this result holds true if instead of iterating a single irrational rotation,
one takes a multi-rotation orbit along a finitely recurrent sequence over
finitely many different irrational rotations. We also discuss some connections
between the box dimensions of multi-rotation orbits and Diophantine
approximations. In particular, we improve a result by Feng and Xiong in the
case when the rotation parameters are algebraic numbers.
