Hausdorff dimension of singular vectors (Q335021)

In this paper, along with many other things, the authors prove two main results regarding the Hausdorff dimension, denoted as \(H_{\dim}\), of singular vectors. To state the results, we need to introduce some notation and terminologies.NEWLINENEWLINEA real vector \((x_1,\ldots, x_d)\in\mathbb R^d\) is singular if, for every \(\varepsilon>0,\) there exists \(T_0\) such that for all \(T>T_0\) the system of inequalities NEWLINE\[NEWLINE\max_{1\leq i\leq d}|qx_i-p_i|<\frac{\varepsilon}{T^{1/d}} \text{ and }0<q<T\tag{1}NEWLINE\]NEWLINE admits an integer solution \((p,q)\in\mathbb Z^d\times \mathbb Z\). Let \(S(d)\) denotes the set of all singular vectors in \(\mathbb R^d\). Clearly, in dimension one, only the rational numbers are singular. The Lebesgue measure of \(S(d)\) is zero, hence, naturally one would like to determine the Hausdorff dimension of this set. It has been proved by the first author that \(S(2)=4/3\) in the paper [Ann. Math. (2) 173, No. 1, 127--167 (2011; Zbl 1241.11075)]. This leads to a question of determining the Hausdorff dimension of \(S(d)\) for \(d\geq 2\).NEWLINENEWLINETheorem 1. For \(d\geq 2\), the Hausdorff dimension of \(S(d)\) is \(\frac{d^2}{d+1}.\)NEWLINENEWLINEThe second main result is about determining the Hausdorff dimension of Dirichlet improvable numbers. Let \(\varepsilon\) be a fixed positive real number. A vector \((x_1,\ldots, x_d)\in\mathbb R^d\) is \(\varepsilon\)-Dirichlet improvable if the system of inequalities (1) admits a solution for \(T\) large enough. Denote the set of \(\varepsilon\)-Dirichlet improvable vectors by \(DI_\varepsilon(d)\).NEWLINENEWLINETheorem 2. Let \(d\geq 2\) be an integer, and let \(t\) be any positive real number greater than \(d\). There is a constant \(C\) such that, for \(\varepsilon\) small enough,NEWLINENEWLINENEWLINE\[NEWLINE\frac{d^2}{d+1}+\varepsilon^t\leq H_{\dim} DI_\varepsilon(d)\leq \frac{d^2}{d+1}+C\varepsilon^{d/2}.NEWLINE\]NEWLINENEWLINENEWLINEThe proofs rely on three main tools, the geometry of numbers, self-similar covering and best Diophantine approximation. The paper is well written, important and may have far-reaching applications. This paper naturally intrigues the reader to know more about the Hausdorff dimension of \(\varepsilon\)-Dirichlet non-improvable vectors or more generally the Hausdorff measure analogues of Theorems 1 and 2.