Suppose I have points uniformly randomly distributed throughout $R^n$, with coordinates restricted to some finite interval centered on the origin. It is then trivial to map those points into the discrete space $Z^n$, e.g. by applying the floor function to each coordinate. Given the assumption of uniformity, each point in the discrete space should then be associated with approximately the same number of source points in the original space.
Now, if we relax the initial assumption of uniformity, we can still achieve a uniform distribution of the number of source points mapped to any given discretized point by using a slightly more complex discretization procedure--rather than simply rounding down to the nearest integer, we can divide up each axis into discrete buckets of varying size, such that each range contains the same number of points whose coordinate values for that axis fall into that range.
Finally, assume that points are not distributed throughout $R^n$, but are in fact restricted to the surface of an (n-1)-sphere. What procedure could be used to map those points, whose distribution is not specified, to a discrete $Z^{n-1}$ space with a uniform distribution of how many source points are mapped to each discrete point?