Hausdorff measure of sets of Dirichlet non-improvable affine forms

2020/06/10 by Taehyeong Kim, Wooyeon Kim, Kim, Taehyeong +1 · 1 citation
Mathematics · #2020: Primary 11J20 #37A17 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Secondary 11K60

paper · pdf · doi:10.48550/arxiv.2006.05727

openalex publication_date 2020/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a decreasing real valued function ψ, a pair (A,b) of a real m× n matrix A and b∈ℝ^m is said to be ψ-Dirichlet improvable if the system ‖Aq+b-p‖^m lt; ψ(T)\quadand ‖q‖ⁿ lt; T has a solution p∈ℤ^m, q∈ℤⁿ for all sufficiently large T, where ‖⋅‖ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the ψ-Dirichlet non-improvable set. In this paper, we prove a similar criterion for the Hausdorff measure of the ψ-Dirichlet non-improvable set. Also, we extend this result to the singly metric case that b is fixed. As an application, we compute the Hausdorff dimension of the set of pairs (A,b) with uniform Diophantine exponents \widehatw(A,b)≤ w.

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