All Santa has to do is go out and deliver presents to some number (M) of houses. Then he shifts over into the next dimension (like in the comics, not like in string theory) and delivers presents for another M houses. Santa does this until he has made deliveries in some number of different dimensions (N) and then he goes home.
For an infinite number of dimensions and Santas in those dimensions if M and N are greater than some number (I suggest it is 2 and 2) then everyone gets presents without Santa even having to break a sweat. I leave it as an exercise to the theorist to show that not all universes have to have the same number of homes.