In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure’s lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain “overlapping” self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers.We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is s-regular.

中文翻译：

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措施和规律性的较低维度

与集合的较低 Assouad 维度类似，我们研究度量的较低 Assouad 维度。与较高的 Assouad 维度一样，度量的较低 Assouad 维度提供有关度量的极端局部行为的信息。我们研究与其他维度和规律性的联系。特别是，准较低的 Assouad 维度由该度量的较低局部维度的下确界支配。虽然严格的不等式通常是可能的，但等式适用于有限类型的自相似测度类。此类包括满足开集条件的所有自相似、等收缩测度，以及某些“重叠”自相似测度，例如具有与皮索数成反比的收缩因子的伯努利卷积。s-常规的。