本文研究了丢番图指数μ_{n，l}$对整数1 \leq l <n和向量α \in \mathbb{R}^n的定义，令μ_{n，l}= \sup{μ\geq 0: 0 < ||x \cdot α|| < H(x)^{-μ} for infinitely many x \in C_{n,l} \cap \mathbb{Z}^n}，其中\cdot是标量积，||. || 表示到最近整数的距离，C_{n,l}是由具有在第一个l坐标中获得的高度的所有矢量组成的广义圆锥。我们证明了指数取区间[l +1，\infty）中的所有值，并且几乎对所有α都取n值.我们计算了μ \geq n中向量集α的Hausdorff维数，其中μ_{n，l}（α）= μ.最后，设w_n表示去掉x的限制后的指数，我们证明了存在向量α使得递增序列μ_{n，1}（α）\leq. \leq μ_{n，n-1}（α）\leq w_n（α）可以任意选取。
英文摘要：
We are studying the Diophantine exponent μ_{n,l}$ defined for integers 1 \leq l < n and a vector α\in \mathbb{R}^n by letting μ_{n,l} = \sup{μ\geq 0: 0 < ||x \cdot α|| < H(x)^{-μ} for infinitely many x \in C_{n,l} \cap \mathbb{Z}^n}, where \cdot is the scalar product and || . || denotes the distance to the nearest integer and C_{n,l} is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1, \infty), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors αwith μ_{n,l} (α) = μfor μ\geq n. Finally, letting w_n denote the exponent obtained by removing the restrictions on x, we show that there are vectors αfor which the gaps in the increasing sequence μ_{n,1} (α) \leq ... \leq μ_{n,n-1} (α) \leq w_n (α) can be chosen to be arbitrary.
Hausdorff measures of sets in Exact Diophantine approximation
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；School of Statistics and Mathematics, Hubei University of Economics（湖北经济学院统计与数学学院）；Hubei Center for Data and Analysis（湖北省数据分析中心）
