In this paper, we consider singular systems of linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular systems of linear forms by constructing an appropriate Margulis height function on the space of lattices over global function fields.

Let.. = (..1,.,....) be a..-tuple of positive real numbers such that S...... = 1 and.. 1........ A..dimensional vector.. = (..1,.,....). R.. is said to be..- singular if for every.. > 0, there exists.. 0 > 1such that for all.. >.. 0, the system of inequalities max 1...... |...... -.... | 1.... <.... and 0 <.. <.. has an integer solution (..,..) = (..1,.,....,..). Z.. x Z. We prove that the Hausdorff dimension of the set of..singular vectors in R.. is bounded below by.. - 1 1+.. 1. Our result partially extends the previous result of Liao et al. [Hausdorff dimension of weighted singular vectors in R2, J. Eur. Math. Soc. 22 (2020), 833-875].

For given & varepsilon;>0 and b is an element of R-m, we say that a real mxn matrix A is & varepsilon;-badly approximable for the target b if lim inf(q is an element of Zn,& Vert;q & Vert;->infinity)& Vert;q & Vert;(n)< Aq-b >(m)>=& varepsilon;, where <& sdot;> denotes the distance from the nearest integral vector. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of & varepsilon;-badly approximable matrices for fixed target b and the set of & varepsilon;-badly approximable targets for fixed matrix A. Moreover, we give a Diophantine condition of A equivalent to the full Hausdorff dimension of the set of & varepsilon;-badly approximable targets for fixed A. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the A-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.

Kurzweil's theorem ('55) is concerned with zero-one laws for well approximable targets in inhomogeneous Diophantine approximation under the badly approximable assumption. In this article, we prove the divergent part of a Kurzweil type theorem via a suitable construction of ubiquitous systems when the badly approximable assumption is relaxed. Moreover, we also discuss some counterparts of Kurzweil's theorem.

A pair (A,𝐛) of a real m× n matrix A and 𝐛∈ℝ^m is said to be infinitely badly approximable if lim inf_𝐪∈ℤ^n, 𝐪→∞𝐪^n/mA𝐪-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.

We study inhomogeneous Diophantine approximation over the completion Kv of a function field K (over a finite field) for a discrete valuation v, with affine algebra Rv. We obtain an effective upper bound for the Hausdorff dimension of the set �BadA(⠂) = & UTheta; E Kvm : � lim inf ⠄q ⠄n ⠄Aq - & UTheta; - p ⠄m ⠂ ⠂ , (p,q)ERvm xRvn,IIqII-& iota;oo of ⠂-badly approximable targets & UTheta; E Kvm given A E Mm,n(Kv), using an effective version of entropy rigidity in homogeneous dynamics for some diagonal action on the space of Rv-grids. We characterize matrices A for which BadA(⠂) has full Hausdorff dimension for some ⠂ > 0 by a Diophantine condition of singularity on average. Our methods work for the approximation using weighted ultrametric distances.& COPY; 2023 Elsevier Inc. All rights reserved.

For a decreasing real valued function 0, a pair (A, b) of a real m x n matrix A and b E Rm is said to be 0-Dirichlet improvable if the system IIAq + b- pII(m) < 0(T) and IIqII(n) < T has a solution p E Z(m), q E Z(n) for all sufficiently large T, where II . II denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the 0-Dirichlet non-improvable set. In this paper, we prove a similar criterion for the Hausdorff measure of the 0-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 w(A,b) <= w. (c) 2022 The Authors. Published by Elsevier Inc.