Gap distribution of √n \,mod\, 1 and the circle method

The distribution of the properly renormalized gaps of √n \,mod\, 1 with $n < N$ converges (when N→ ∞) to a non-standard limit distribution, as Elkies and McMullen proved in 2004 using techniques from homogeneous dynamics. In this paper we give an essentially self-contained proof based on the circle method. Our main innovation consists in showing that a new type of correlation functions of √n \,mod\, 1 converge. To define these correlation functions we restrict, smoothly, to those √n \,mod\, 1 that lie in minor arcs, i.e. away from rational numbers with small denominators.