We show that if $X$ is a uniformly perfect complete metric space satisfyingthe finite doubling property, then there exists a fully supported measure withlower regularity dimension as close to the lower dimension of $X$ as we wish.Furthermore, we show that, under the condensation open set condition, the lowerdimension of an inhomogeneous self-similar set $E_C$ coincides with the lowerdimension of the condensation set $C$, while the Assouad dimension of $E_C$ isthe maximum of the Assouad dimensions of the corresponding self-similar set $E$and the condensation set $C$. If the Assouad dimension of $C$ is strictlysmaller than the Assouad dimension of $E$, then the upper regularity dimensionof any measure supported on $E_C$ is strictly larger than the Assouad dimensionof $E_C$. Surprisingly, the corresponding statement for the lower regularitydimension fails.
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机译：我们证明如果$ X $是满足有限加倍属性的一致完美的完全度量空间，则存在一个完全支持的度量，其正则性维数较低，与我们希望的近似于$ X $的维数较小。在凝结开放集条件下，不均匀的自相似集$ E_C $的较低维度与凝结集$ C $的较低维度重合，而$ E_C $的Assouad维度是相应自相似的Assouad维度的最大值设置$ E $，冷凝设置$ C $。如果$ C $的Assouad维严格小于$ E $的Assouad维，则$ E_C $支持的任何度量的较高正则性维都严格大于$ E_C $维的Assouad维。令人惊讶的是，较低的正则性维度的相应陈述失败了。
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