2022/06/10 by Uri Shapira, Shapira, Uri, Barak Weiss +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2206.05329
openalex publication_date 2022/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let θ∈ℝd. We associate three objects to each approximation (p,q)∈ ℤd× ℕ of θ: the projection of the lattice ℤd+1 to the hyperplane of the first d coordinates along the approximating vector (p,q); the displacement vector (p - qθ); and the residue classes of the components of the (d + 1)-tuple (p, q) modulo all primes. All of these have been studied in connection with Diophantine approximation problems. We consider the asymptotic distribution of all of these quantities, properly rescaled, as (p, q) ranges over the best approximants and ε-approximants of θ, and describe limiting measures on the relevant spaces, which hold for Lebesgue a.e. θ. We also consider a similar problem for vectors θ whose components, together with 1, span a totally real number field of degree d+1. Our technique involve recasting the problem as an equidistribution problem for a cross-section of a one-parameter flow on an adelic space, which is a fibration over the space of (d + 1)-dimensional lattices. Our results generalize results of many previous authors, to higher dimensions and to joint equidistribution.