This article presents a novel singularity concept for affine forms, characterizing infinitely badly approximable pairs and tackling their Hausdorff dimension.
A pair (A,b) of a real m× n matrix A and b∈ Rᵐ is said to be infinitely badly approximable if align* & _qⁿ, \|q\|→∞ \|q\|n/m\|Aq-b\|Z =∞, align*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 Rᵐ⁺ⁿ.
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Taehyeong Kim (2025) studied this question.
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