Several suggested a set of points 1 mile north of the South Pole. This is not a legitimate answer, because once you arrive at the South Pole, you can not proceed to "walk" eastwardly. Any forward motion whatsoever is, by definition, northward. And I just can't accept stationary spinning as a form of walking.
However, that is very close to the answer, both in concept and in location. Back up an infinitesimal distance further north (slightly more than a mile away from the South Pole). You now have a "ring" around the globe, an infinite set of points all slightly more than a mile away from the South Pole. From any one of these points, following the southern directed 1 mile trek, you will end up just north of the South Pole. You now *can* walk east for 1 mile. You will complete a LOT of tiny circles around the South Pole. If your rings of starting points was located at the proper distance away, you will complete the 1 mile eastward trek exactly where you started that leg of the journey (completing exactly "N" circles). And then you can follow your path in the snow (north this time) back to the exact starting point!
But wait, that's not all. Let's start over. This time start from another ring that is slightly further north. Just the right distance away from the South Pole so that your eastward 1 mile trek will complete exactly "n-1" complete circles. Now you have found 2 rings, each with an infinite number of points.
As you might imagine, you can keep backing up further north, creating add'l starting point rings, until you are about 1.16 miles away from the South Pole, at which point, following your 1 mile southward trek, your eastward leg will complete only 1 full circle. At this point, you will have found "N" rings.
So, you will have N rings starting just slightly more than 1.00 miles away from the South Pole, and ending at about 1.16 miles away.
N = infinity, since the closer you get to 1.00 miles away will work, as long as you can complete exactly "n" circles during the eastward trek. As you can see, there are actually "I^2+1" points that work, where I = infinity. The "+1" brings in the one point in the Northern hemisphere that also works, which is, of course, the North Pole (pick either true or magnetic north).
To view the puzzle click here