## Hausdorff dimension of singular vectors.(English)Zbl 1358.11078

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.
A 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 $\max_{1\leq i\leq d}|qx_i-p_i|<\frac{\varepsilon}{T^{1/d}} \text{ and }0<q<T\tag{1}$ 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 $$d\geq 2$$.
Theorem 1. For $$d\geq 2$$, the Hausdorff dimension of $$S(d)$$ is $$\frac{d^2}{d+1}.$$
The 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 $$DI_\varepsilon(d)$$.
Theorem 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,
$\frac{d^2}{d+1}+\varepsilon^t\leq H_{\dim} DI_\varepsilon(d)\leq \frac{d^2}{d+1}+C\varepsilon^{d/2}.$
The 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.

### MSC:

 11J13 Simultaneous homogeneous approximation, linear forms 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 11K60 Diophantine approximation in probabilistic number theory 37A17 Homogeneous flows

Zbl 1241.11075
Full Text: