regarding the nr 1: the reason I'd prefer solving the rectangular version first is because I see 2 potential sources of problem: on one hand an anisotropic but still
flat metric (i.e. no curvature) and on the other hand the shape or topology of spaces like sphere (which admits elliptic metrics) or double torus (hyperbolic geometry) which force curvature on the metric. That the problem arises already for a flat metric is why I think it deserves attention.
Although the problem can be seen with visual inspection, it incurs some effort to inspect. I propose the following visualization: after generating say N=1000 points, for each point select the closest neighbour, and plot its distance with respect to the origin, then ideally one would get a circular band centered on the origin. Alternatively, plot the angle for each nearest neighbour vector and ideally one should see a uniform distribution.
It's unclear which part(s) need to be generalized for the rectangular version: the equation for the phi's? some rescaling of the phi's? the update rule itself? or some function on the output of the update rule?
Another approach may be to change the concept of hyper uniformity and generalize it to "hyper convergence of [arbitrary function]" i.e. can we have fast convergence to a hat or tent function? If we take the 2-dimensional R_2 and compute the sum of the pair of pseudo-independent numbers, does it generate a tent function? still faster than random sequence? Instead of looking at the distance between adjacent points for the non-uniform tent function, we might multiply this distance by the distance to 0 or 2 (whichever is closest). I.e. points near the floor (0 or 2) should be further apart but will have a lower multiplier since its close to 0 or 2, while points near the top of the tent function (1) will have small distances between them but a higher distance fromm 0 or 2.
For a couple of moments I incorrectly conjectured that spaces have "magic geodesic steps" i.e. for a torus moving straight up or straight to the right does not create a dense point set on the unit square, while the R_2 step does. The reason I think there is no general "magic geodesic step" or at least that it is less related to geodesics as I thought is because every geodesic on the unit sphere turns back on itself.