We derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel–Cantelli-type arguments.

中文翻译：

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随机自仿射地毯的 Assouad 谱

我们推导出单变量随机自仿射 Bedford-McMullen 地毯的几乎确定的 Assouad 谱和准 Assouad 维数。以前的工作表明，（相关的）Assouad 维度不足以区分随机模型中的细微变化，因为它几乎可以肯定地“尽可能大”（一个确定性的量）。这已在保形和非保形设置中得到验证。在保形设置中，Assouad 谱和准 Assouad 维的行为相当不同，几乎可以肯定地与上框维重合。在这里，我们研究了非共形设置，发现 Assouad 谱和准 Assouad 维数通常与 box 维数或 Assouad 维数不重合。我们提供的例子突出了这些概念之间的细微差别。