广义中间维数
Generalised intermediate dimensions
AI总结：
本文提出了介于Hausdorff与box维数之间的$Φ$-中间维数，通过限制覆盖集相对大小实现更精细的几何刻画，并将其推广至更广度量空间，证明了插值恢复、连续性下界及Hölder畸变等结果。
摘要：
我们引入了一族维数，称为$Φ$-中间维数，它们介于Hausdorff维数与box维数之间，并推广了Falconer、Fraser和Kempton引入的中间维数。这是通过限制覆盖集的相对大小来实现的，该方式比中间维数的定义允许更精细的细化。我们还将理论从欧几里得空间扩展到更广泛的度量空间类。我们证明，对于在$θ=0$处中间维数不连续的紧致子集，这些维数可用于“恢复”其Hausdorff维数与box维数之间的插值，从而提供关于此类集合更精细的几何信息。我们证明了涉及Assouad维数和下维数的连续性类结果，给出了中间维数的一个精确的一般下界，对于具有正box维数的集合，该下界对所有$θ\in (0,1]$均为正。我们还证明了Hölder畸变估计、质量分布原理和Frostman型引理，并用它们来研究乘积集合的维数。
英文摘要：
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.
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