可达到的几乎 surely 大维度
The attainable almost sure large dimensions
AI总结：
本文研究了由随机Moran测度自然模型生成的随机测度的几乎 surely 维度范围，探讨了特定尺度深度范围内的局部几何信息，并确定了随机Moran集上随机测度的几乎 surely 维度。
摘要：
在本文中，我们研究了由自然随机Moran测度模型生成的随机测度的可能几乎 surely 维度范围。具体而言，我们考虑这些测度的Assouad-like ``大'' Φ-维度。这些维度可以调节以考虑特定尺度深度范围，从而提供更精细的局部几何信息。准Assouad维度是已知且重要的``大'' Φ-维度的例子。我们确定了由该模型生成并支持于任意给定随机Moran集的随机测度的可能几乎 surely Φ-维度范围。我们分别讨论了当概率权重依赖于缩放因子和不依赖于缩放因子的情况。在后一种情况下，我们显示通常存在集合维度与最小可达到上维度和最大可达到下维度之间的``间隙''。作为我们结果的后果，我们还确定了底层随机Moran集的几乎 surely 维度。
英文摘要：
In this paper we study the range of possible almost sure dimensions of random measures arising from a natural model of random Moran measures. Specifically, we consider the Assouad-like ``large'' $Φ$-dimensions of these measures. These dimensions can be tuned to consider a specific range of depths in scale and so provide refined local geometric information. The quasi-Assouad dimension is a well-known and important example of a ``large'' $Φ$-dimension. We determine the range of possible almost sure $Φ$-dimensions for random measures generated by the model and supported on any given random Moran set. We do this for both the case when the probability weights depend on the scaling factors and the case when they do not. In the later situation, we show that usually there is a ``gap'' between the dimension of the set and that of the smallest attainable upper dimension and largest attainable lower dimension. As a consequence of our results, we also determine the a.s. dimensions of the underlying random Moran set.
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