We show that if X is a uniformly perfect complete metric space satisfying
the finite doubling property, then there exists a fully supported measure with lower regularity
dimension as close to the lower dimension of X as we wish. Furthermore, we show that, under
the condensation open set condition, the lower dimension of an inhomogeneous self-similar set EC
coincides with the lower dimension of the condensation set C, while the Assouad dimension of
EC is the maximum of the Assouad dimensions of the corresponding self-similar set E and the
condensation set C. If the Assouad dimension of C is strictly smaller than the Assouad dimension
of E, then the upper regularity dimension of any measure supported on EC is strictly larger than
the Assouad dimension of EC. Surprisingly, the corresponding statement for the lower regularity
dimension fails.
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