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.)

[![enter image description here][1]][1]

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. 


  [1]: https://i.sstatic.net/JkvjA.png