Twisted Diophantine approximation for matrix transformations of tori

TL;DR

This paper investigates the size and measure-theoretic properties of sets of vectors approximated by matrix transformations of tori, establishing a dichotomy and a Jarník-type theorem under various conditions.

Contribution

It proves a metric dichotomy for the set of approximated vectors under mild conditions, extending to Lebesgue measure and solving a conjecture in the field.

Findings

Establishes a metric dichotomy for almost every vector with respect to fractal measures.

Proves a Lebesgue measure dichotomy without restrictions on the approximation functions.

Provides a Jarník-type theorem for the approximation sets.

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 ${\bm \alpha},$ we are interested in the set $\mathcal{T}_{{\bm \alpha}}(\mathcal{A}, {\bf r})$ of vectors ${\bm \beta}\in[0,1)^{d}$ for which $A_n{\bm \alpha}~~\!\!\!\!\!\pmod{1}$ infinitely often lies in the box centred at ${\bm \beta}$, 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 $\mathcal{T}_{{\bm \alpha}}(\mathcal{A}, {\bf r}),$ valid for almost every ${\bm \alpha}$ 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 ${\bm \alpha}.$ This solves a variant of a conjecture of Gonz\'{a}lez Robert, Hussain, Shulga and Ward [Conjecture 1.10, Bull. London Math. Soc. 2025]. Finally, we also establish a Jarn\'{i}k-type theorem for $\mathcal{T}_{{\bm \alpha}}(\mathcal{A}, {\bf r}).$