论文标题
广义中间维度
Generalised intermediate dimensions
论文作者
论文摘要
我们介绍了一个尺寸系列，我们称之为$φ$ - 室内尺寸，位于Hausdorff和Box尺寸之间，并推广Falconer，Fraser和Kempton引入的中间维度。这是通过限制覆盖集的相对大小的方式来完成的，该方式允许与中间维度的定义相比，可以进行更大的改进。我们还将理论从欧几里得空间扩展到更广泛的度量空间。我们证明，这些维度可用于在hausdorff和紧凑型子集的盒子尺寸之间“恢复插值”，在$θ= 0 $时中间尺寸不连续，从而提供了有关此类集合的几何信息。我们证明了涉及Assouad和较低维度的类似连续性结果，这为中间维度提供了急剧的一般下限，对于具有正箱尺寸的套装的所有$θ\ in（0,1] $均为正面。我们还证明，我们还证明Hölder扭曲估计值，质量分配原则，一种质量分配原则，以及一种糖果型lemma，我们用于研究产品的diDemensions dimensions dimensions dimensions dimensions dimensions dimensions dimensions dimensions dimensions dimensions dimensions dimensions dimensions dimensions。
We introduce a family of dimensions, which we call the $Φ$-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by restricting the relative sizes of the covering sets in a way that allows for greater refinement than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We prove that these dimensions can be used to 'recover the interpolation' between the Hausdorff and box dimensions of compact subsets for which the intermediate dimensions are discontinuous at $θ=0$, thus providing finer geometric information about such sets. We prove continuity-like results involving the Assouad and lower dimensions, which give a sharp general lower bound for the intermediate dimensions that is positive for all $θ\in (0,1]$ for sets with positive box dimension. We also prove Hölder distortion estimates, a mass distribution principle, and a Frostman type lemma, which we use to study dimensions of product sets.
