Title: Gap distribution of √n modulo 1 and the circle method
Abstract: Let α be a positive non-integer real number. We consider the values n^{α} modulo 1 with n < N and place them in order on the interval [0,1]. It is tempting to conjecture that these values should look like a set of random points, therefore the gap distribution (at the scale 1/N) should be Poisson.
It turns out however, as proven by Elkies and McMullen, that the gap distribution exists and is not Poisson when α = 1/2. This is the only exponent for which the existence of the gap distribution is proven. The proof of Elkies-McMullen and the later proof of Browning-Vinogradov are dynamical in nature. We will present a self-contained and elementary proof based on the circle method.
I will discuss the differences between the two proofs, time permitting. Joint work with Niclas Technau.
