I have plotted solutions $(a,b,c)$ to $a^2 + b^2 + c^2 = n$ for $12000 \leq n < 12100$, rescaled to $S^2$ and projected onto the first two coordinates. (these are read from the lower left, across and upwards. Sorry.)

While, over all $ a \leq n \leq b$ there is clear tendency towards equidistribution, for a fixed $n$, there are marked patterns (or even no solution at all). I've been struggling to find language for the types of patterns that are observing. There are solution-free regions, there are linear patterns, circular patterns and other sub-varieties.

I suspect for fixed $n$, the solutions are clustering along the intersection of two vareties, $S^2 \cap V $ and I am trying to characterize the equations of $V$.

This is my theory of why equidistribution might be so hard to prove; it's because there are in fact patterns.