In reply to Rick Wicklin.

Thanks for your response. I mentioned the (non-uniform) marginal distributions because you wrote: "You can use histograms to demonstrate that each coordinate has a uniform marginal distribution." Maybe I have misunderstood this sentence.

My first issue is not about the random gaps and clusters of the empirical distribution, but really about the theoretical distribution. It helped me to envision a uniform distribution on a square (rather than a circle), e.g., S={(x, y) in [-1, 1]²: |x|=1 or |y|=1}. Now, if the linear transformation Y = diag(1, 0.5)*X is applied to this square, isn't the probability density on the vertical sides (length=1) of the resulting rectangle twice as large as on the horizontal sides (length=2)? A similar effect should occur when a circle is transformed to an ellipse: The transformation with diag(1, 0.5) -- or even diag(1, 0.001) -- should lead to an increased probability density near the left and right vertices of the resulting ellipse (in comparison to the co-vertices), hence a non-uniform distribution if the distribution on the circle was uniform.