non-randomness

A question sent by Eric, who attended my Public Lecture last week in Brisbane:

Last year at this time, Peter Sarnak toured Australia talking about randomness in number theory and Moebius randomness in dynamics. Recently,  he pointed a paper on the arXiv in which he claims that the distribution of [integer coordinate] points on the sphere of radius √n which satisfy

is random as n goes to infinity (the paper is much more precise).  You mentioned tests which look for non-randomness.  How does one test for a non-random distribution of points on the sphere?

Interesting question, both for linking two AMSI Lecture tours (Peter Sarnak’s schedule sounded more gruelling than mine!) and for letting me get a look at this paper. Plus for the connection with probabilistic number theory. This paper indeed stands within the area of randomness in number theory rather than random generation and I do not see an obvious connection here, but the authors of the paper undertake a study of the randomness of the solutions to the above equation for a fixed n using statistics and their limiting distribution. (I am not certain of the way points are obtained over a square on Fig. 1, presumably this is using the spherical coordinates of the projections over the unit sphere in R³.) Their statistics are the electrostatic energy, Ripley’s point pair statistic, the nearest neighbour spacing measure, minimum spacing, and the covering radius. The most surprising feature of this study is that this randomness seems to be specific to the dimension 3 case: when increasing the number of terms in the above equation, the distribution of the solutions seems more rigid and less random…