发表机构
School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义了有界集的下、上盒维数印记，用加权覆盖数刻画，证明下印记闭包由偏心率轮廓决定，并研究了其射影不变性及与Hausdorff维数印记的关系。
AI 中文摘要
我们研究了\\(\mathbb R^n\\)中有界子集的下、上盒维数印记，这些印记由具有规定有序边长界的独立定向矩形盒的加权覆盖数定义。极限范围涵盖所有偏心率，包括无界纵横比。对于每个非空有界集，我们在没有任何一致性假设的情况下，将下印记的闭包与由其下偏心率轮廓确定的一半空间的交集等同起来。该轮廓等于此闭包的支撑函数当且仅当它是次可加的。一个平面乘积例子在每个射线上具有不同的下、上轮廓，并且其下印记闭包可显式计算。我们还证明了在避开极点超平面的紧集上，两个印记在非奇异射影变换下不变。对于类型为\\((1,\ldots,n)\\)的非退化曲线、其Ahlfors正则参数子集以及高维球面，一致的各向异性覆盖估计确定了两个印记，包括其边界点。最后，局部覆盖计数和乘积测度准则将下盒印记与Hausdorff维数印记等同起来，而一个倒数序列例子表明这种包含关系可能是严格的。
英文摘要
We study lower and upper box dimension prints for bounded subsets of \(\mathbb R^n\), defined by weighted covering numbers for independently oriented rectangular boxes with prescribed ordered side-length bounds. The limits range over all eccentricities, including unbounded aspect ratios. For every non-empty bounded set, we identify the closure of the lower print with the intersection of the half-spaces determined by its lower eccentricity profile, without any uniformity assumption. The profile equals the support function of this closure if and only if it is subadditive. A planar product example has distinct lower and upper profiles on every ray and an explicitly computable lower-print closure. We also prove that both prints are invariant under nonsingular projective transformations on compact sets avoiding the pole hyperplane. Uniform anisotropic covering estimates determine both prints, including their boundary points, for non-degenerate curves of type \((1,\ldots,n)\), their Ahlfors regular parameter subsets, and higher-dimensional spheres. Finally, local covering-count and product-measure criteria identify the lower box print with the Hausdorff dimension print, while a reciprocal-sequence example shows that this inclusion can be strict.
