LOWER ASSOUAD-TYPE DIMENSIONS OF UNIFORMLY PERFECT SETS IN DOUBLING METRIC SPACES

Abstract

In this paper, we are concerned with the relationship among the lower Assouad-type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as [Formula: see text] tends to 1 is equal to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as [Formula: see text] tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi-lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi-lower Assouad dimension.