Infinitely badly approximable affine forms

2024/06/02 by Tae-Hyeong Kim, Kim, Taehyeong
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.00821

openalex publication_date 2024/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A pair (A,b) of a real m× n matrix A and b∈ℝ^m is said to be infinitely badly approximable if \liminf_{q∈ℤⁿ, ‖q‖→∞} ‖q‖^(n)/(m)‖Aq-b‖_ℤ =∞, where ‖⋅‖_ℤ denotes the distance from the nearest integer vector. In this article, we introduce a novel concept of singularity for (A,b) and characterize the infinitely badly approximable property by this singular property. As an application, we compute the Hausdorff dimension of the infinitely badly approximable set. We also discuss dynamical interpretations on the space of grids in ℝ^m+n.

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