2024/03/25 by Maksym Radziwiłł, Radziwiłł, Maksym, Niclas Technau +1
Mathematics · #11J71 #37A44 #65C20 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical functions and polynomials #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2403.16493
openalex publication_date 2024/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.