Twisted Diophantine approximation for matrix transformations of tori

TL;DR

This work advances twisted Diophantine approximation on tori driven by expanding integer matrices by proving a Khintchine-type metric dichotomy for the target set $\mathcal{T}_{\boldsymbol{\alpha}}(\mathcal{A},\mathbf{r})$ under Fourier decay, and a Lebesgue-almost-everywhere version without the doubling assumption on $\mathbf{r}$. It introduces a dynamical, matrix-parameterized framework for twisted approximation and establishes a Jarník-type theorem for the corresponding Hausdorff measures. The proofs blend dynamical equidistribution, Fourier-analytic decay, Borel–Cantelli machinery, and mass transference principles to connect the arithmetic structure of $A_n$ with metric size properties of the limsup sets. Applications include a variant of a conjecture by González Robert et al. and sharpened descriptions of the fine-scale size (Hausdorff dimension and measure) of twisted approximation sets in toral dynamics. Overall, the results unify fractal-measure dichotomies with dynamical toral endomorphisms, yielding precise thresholds for size across measure-theoretic and fractal scales.

Abstract

Consider a sequence of integral matrices $\mathcal{A}=(A_n)_{n\in\N}$, and a $d$-tuple function ${\bf r}=(r_1,\ldots,r_d)\colon \N\to (0,\frac{1}{2})$. For a fixed vector ${\bm α},$ we are interested in the set $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r})$ of vectors ${\bm β}\in[0,1)^{d}$ for which $A_n{\bm α}~~\!\!\!\!\!\pmod{1}$ infinitely often lies in the box centred at ${\bm β}$, with side lengths $2r_i(n)$ in each coordinate direction. Under mild conditions on $\mathcal{A}$ and ${\bf r}$, we prove a metric dichotomy for the size of $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r}),$ valid for almost every ${\bm α}$ with respect to any fractal measure with a certain polynomial Fourier decay rate. Furthermore, removing all restrictions on ${\bf r}$, we establish a metric dichotomy for Lebesgue almost every ${\bm α}.$ This solves a variant of a conjecture of González Robert, Hussain, Shulga and Ward [Conjecture 1.10, Bull. London Math. Soc. 2025]. Finally, we also establish a Jarník-type theorem for $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r}).$

Theorems & Definitions (31)

• Theorem 1: Kurzweil Kur55
• Theorem 2: B03SchmTrot03
• Theorem 3: TZ25
• Theorem 4
• Example 5
• Proposition 6
• Corollary 7
• Theorem 8
• Remark 9
• Conjecture 10: RHSW25