On a lower bound of Hausdorff dimension of weighted singular vectors (Q6581855)

The notion of singularity in the sense of Diophantine approximation was introduced by \textit{A. Khintchine} in [Rend. Circ. Mat. Palermo 50, 170--195 (1926; JFM 52.0183.01)] in 1926. That is, it is noted that \(d\)-dimensional vector \(x=(x_1, \dots, x_d) \in \mathbb{R}^d\) is said to be \(w\)-singular if for every \(\epsilon >0\), there exists \(T_0>1\) such that for all \(T>T_0\), the system of inequalities \N\[ \N\max_{1 \leqslant i \leqslant d} |qx_i - p_i|^{\frac{1}{w_i}} < \frac{\epsilon}{T} \quad \text{and} \quad 0<q<T \N\] \Nhas an integer solution \((\mathbf{p}, q)=(p_1, \dots, p_d, q) \in \mathbb{Z}^d \times \mathbb{Z}\). In this paper, \(w=(w_1, \dots, w_d)\) is a \(d\)-tuple of positive real numbers such that \(\sum_i w_i =1\) and \(w_1 \geqslant \cdots \geqslant w_d\).\N\NThe main result is to show that ``the Hausdorff dimension of the set of \(w\)-singular vectors in \(\mathbb R^d\) is bounded below by \(d-\frac{1}{1+w_1}\)''. The authors note that this result partially extends the previous result of the paper [\textit{L. Liao} et al., J. Eur. Math. Soc. (JEMS) 22, No. 3, 833--875 (2020; Zbl 1433.11084)].\N\NThe notion of the singularity is explained. Investigations on the Hausdorff dimension of the sets of singular matrices, are briefly described. The special attention is also given to a discussion on fractal and self-affine structures.