  We study recurrence, and multiple recurrence, properties along the $k$-th
powers of a given set of integers. We show that the property of recurrence for
some given values of $k$ does not give any constraint on the recurrence for the
other powers. This is motivated by similar results in number theory concerning
additive basis of natural numbers. Moreover, motivated by a result of Kamae and
Mend\`es-France, that links single recurrence with uniform distribution
properties of sequences, we look for an analogous result dealing with higher
order recurrence and make a related conjecture.
