M ar 2 01 2 LOCAL STATISTICS OF LATTICE POINTS ON THE SPHERE

The set of integer solutions (x 1 , x 2 , x 3) to the equation (1.1) x 2 1 + x 2 2 + x 2 3 = n has been much studied. However it appears that the spatial distribution of these solutions at small and critical scales as n → ∞ have not been addressed. The main results announced below give strong evidence to the thesis that the solutions behave randomly. This is in sharp contrast to what happens with sums of two or four or more squares. First we clarify what we mean by random. For a homogeneous space like the k-dimensional sphere S k with its rotation-invariant probability measure σ, the binomial process is what you get by placing N points P 1 ,. .. , P N on S k independently according to σ. We are in interested in statistics, that is functions f (P 1 ,. .. , P N), which have a given behaviour almost surely, as N → ∞. If this happens we say that this behaviour of f is that of random points. We shall also contrast features of random points sets with those of " rigid " configurations, by which we mean points on a planar lattice, such as the honeycomb lattice.