Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions in this question). The following questions were asked by j.c. in comments:

Are the points in the Brillouin zone become equidistributed on the circle? Is it possible to say anything about the asymptotics of the "outer perimeter" of the Brillouin zone as well?

And one more question:

Is the area of $N$th Brillouin zone equidistributed along the corresponding circle?