Ramírez's problems and fibers on well approximable set of systems of affine forms

TL;DR

This work develops a comprehensive fibered framework for inhomogeneous Diophantine approximation of $m\times n$ affine systems, identifying when badly approximable matrices fail to be approximated at monotone rates and detailing the structure of well- and badly-approximable fibers. By defining and analyzing sets such as $W_{m,n}(\psi)$, $\Omega(m,n)$, $\Lambda(m,n)$ and their fibers, the authors resolve Ramírez's Kurzweil-type questions in many regimes (notably $mn>1$), establish precise Hausdorff-dimension results, and reveal rich Baire-category topologies for approximation-function families. They demonstrate duality between approximation and badly-approximable sets via $\Lambda(m,n)$ and $\boldsymbol{\rm Bad}(m,n)$, showing full-measure fibers and sharp dimension bounds for the corresponding bad sets. The results advance understanding of inhomogeneous Kurzweil-type phenomena, fibered approximation problems, and the topological landscape of monotone-rate approximation functions.

Abstract

We show that badly approximable matrices are exactly those that, for any inhomogeneous parameter, can not be inhomogeneous approximated at every monotone divergent rate, which generalizes Ramírez's result (2018). We also establish some metrical results of the fibers on well approximable set of systems of affine forms, which gives answer to two of Ramírez's problems (2018). Furthermore, we prove that badly approximable systems are exactly those that, can not be approximated at each monotone convergent rate ψ. Moreover, we study the topological structure of the set of approximation functions.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Corollary 1.6
• Remark 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10