A theorem of Gauss and Legendre states that every integer n that is not congruent to 0,4 or 7 mod 8 can be presented as a sum of three relatively prime squares. In fact, we know (ineffectively) that under these conditions there are n^{1/2+o(1)} such representations of n.

 

Rescaling the points to the unit sphere, we get as n increases more and more points on this sphere, and some 50 years ago Linnik, in a book with the intriguing name "Ergodic properties of number fields", used ergodic theory to study the distribution of these points, at least for sequences of n satisfying an auxiliary congruence condition --- for instance if n is in addition assumed to be congruent to 2 (mod 3). 

 

An alternative route, using bounds on Fourier coefficients of certain automorphic forms, was given by Duke and Iwaniec. This alternative route has several advantages, e.g. does not need a congruence condition and gives much more quantitative information. But the highly original work of Linnik on the subject is still very relevant, and had motivated much recent work.

 

I will explain why and how ergodic theory --- more specifically, homogeneous dynamics --- can be used to study integer points on the sphere, and time permitting present some recent work in this direction.