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 θ tends to 1 equals 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 θ 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.

Lower Assouad Dimension of Measures and Regularity

“…We also give an example to show that equality of the upper and lower Assouad dimensions does not imply s-regularity of the measure. In analogy with what was shown for sets in [2] and [4], we prove that the quasi-lower and quasi-upper Assouad dimensions of measures can be recovered from the Assouad dimension spectrum of a measure under the assumption that the measure is quasidoubling, i.e., has finite quasi-upper Assouad dimension. These results can all be found in Sections 2 and 6.…”

“…The corresponding result was later proved for the quasi-lower Assouad dimension in [2] (with the additional assumption that the space E was uniformly perfect). It is straightforward to obtain the analogous result for doubling measures, that is measures µ for which dim A µ < ∞.…”

“…In [2] it was shown that dim ≤θ A := h(1/θ − 1) = inf 0≤ψ≤θ dim =ψ A under the assumption that the metric space is doubling and uniformly perfect. In this section we will give a shorter proof of this fact that does not require the uniformly perfect assumption.…”

“…. , 2 n , so I (1) n , I (2) n are the two descendants of I (1) n−1 , for example. Let {n j } be an integer sequence with n j+1 ≥ 3n j .…”

“…These results can all be found in Sections 2 and 6. In the appendix, we simplify the proof given in [2] that the quasi-lower Assouad dimension of a doubling metric space is the limit of the dimension spectrum and remove their assumption that the metric space is uniformly perfect.…”