  Consider the integer points lying on the sphere of fixed radius projected
onto the unit sphere. Duke showed that, on congruence conditions for the radius
squared, these points equidistribute. To further this study of
equidistribution, we consider the variance of the number of points in a
spherical cap. An asymptotic for this variance was conjectured by
Bourgain-Rudnick-Sarnak. We prove an upper bound of the correct size on the
average (over radii) of these variances.
