Gap distribution of $\sqrt{n}\mathrm{m}\mathrm{o}\mathrm{d}1$ and the circle method

The distribution of the properly renormalized gaps of $\sqrt{n}\mathrm{m}\mathrm{o}\mathrm{d}1$ with $n<N$ converges (when $N\to \mathrm{\infty }$) 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 $\sqrt{n}\mathrm{m}\mathrm{o}\mathrm{d}1$ converge. To define these correlation functions we restrict, smoothly, to those $\sqrt{n}\mathrm{m}\mathrm{o}\mathrm{d}1$ that lie in minor arcs, i.e. away from rational numbers with small denominators.