Dimension theory of inhomogeneous Diophantine approximation with matrix sequences

TL;DR

This paper studies the Hausdorff dimension of inhomogeneous well-approximable points defined by matrix sequences, unifying shrinking target and recurrence sets, and providing explicit formulas and bounds.

Contribution

It introduces new bounds and formulas for Hausdorff dimensions in inhomogeneous Diophantine approximation with matrix sequences, extending existing principles and applying to concrete matrix classes.

Findings

Derived bounds involve singular values and lattice minima.

Unified shrinking target and recurrence set analysis.

Extended Mass Transference Principle for rectangles.

Abstract

In this paper, we investigate the Hausdorff dimension of naturally occurring sets of inhomogeneous well-approximable points with a sequence of real invertible matrices $\mathcal{A}=(A_n)_{n\in\mathbb{N}}$. Specifically, for a given point $\mathbf{y}\in [0,1)^d$ and a function $\psi : \mathbb{N} \to \mathbb{R}^+$, we study the limsup set \[ W\big(\mathcal{A},\psi,{\bf y}\big) =\Big\{\mathbf{x}\in [0,1)^d\colon A_n\mathbf{x}~(\bmod~1)\in B\big(\mathbf{y}, \psi(n)\big) {\rm ~ for~ infinitely ~many}~n\in\mathbb{N}\Big\}.\] The upper and lower bounds on the Hausdorff dimension of $W\big(\mathcal{A},\psi,{\bf y}\big)$ are determined by involving the singular values of $A_n$ and the successive minima of the lattice $A_n^{-1}\mathbb{Z}^d$, and both bounds are shown to be attainable for some matrices. Within this framework, we unify the problem of shrinking target sets and recurrence sets, establishing the Hausdorff dimensions for such limsup sets. As applications, our corresponding upper bounds for shrinking target and recurrence sets essentially improve those appearing in the present literature. Furthermore, explicit Hausdorff dimension formulas are derived for shrinking targets and recurrence sets associated with concrete classes of matrices. We extend the Mass Transference Principle for rectangles of Li-Liao-Velani-Wang-Zorin (Adv. Math., 2025) to rectangles under local isometries. This generalization yields a general lower bound for the Hausdorff dimension of $W\big(\mathcal{A},\psi,{\bf y}\big)$.