2024/03/25 by Woo-Yeon Kim, Kim, Wooyeon
Computer Science · #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2403.16559
openalex publication_date 2024/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For d≥ 3 we first show that the Hausdorff dimension of the set of A-divergent on average points in the (d-1)-dimensional closed horosphere in the space of d-dimensional Euclidean lattices, where A is the group of positive diagonal matrices, is at most (d-1)/(2). In particular, this upper bound is sharp for d=3. We apply this to compute the Hausdorff dimension of the set of exceptions to the inhomogeneous uniform version of Littlewood conjecture. We say that a pair (ξ1,ξ2)∈ℝ2 satisfies the inhomogeneous Littlewood conjecture if \liminfq→∞q‖qξ1-θ1‖ℤ‖qξ2-θ2‖ℤ=0 for all (θ1,θ2)∈ℝ2, where ‖⋅‖_ℤ denotes the distance to the nearest integer. We prove that the Hausdorff dimension of the set of pairs (ξ1,ξ2)∈ℝ2 not satisfying the inhomogeneous Littlewood conjecture is 1, which is equal to the Hausdorff dimension of the conjectural set of exceptions.