Let E = E({nk},{ck}) be a fat uniform Cantor set. We prove that E is a minimally fat set for doubling measures if and only if (nkck)p = ∞ for all p < 1 and that E is a fairly fat set for doubling measures if and only if there are constants 0 < p < q < 1 such that (nkck)q < ∞ and (nkck)p = ∞. The classes of minimally thin uniform Cantor sets and of fairly thin uniform Cantor sets are also characterized.
In this paper, we construct a scattered Cantor set having the value 1/2 of log2/log3- dimensional Hausdorff measure. Combining a theorem of Lee and Baek, we can see the value 21 is the minimal Hausdorff measure of the scattered Cantor sets, and our result solves a conjecture of Lee and Baek.
让 E 一个饼干切割器被放与暗淡 H E = s。Hausdorff s 措施和集合 E 的收拾行李的 s 措施积极、有限，这被知道。在这篇论文，我们证明为计量器功能 g，集合 E 有积极、有限的 Hausdorff g 措施如果并且仅当 0 <
lim inf t→0 <
∞
。另外，如果并且仅当，我们证明为加倍的计量器功能 g，集合 E 有积极、有限的收拾行李的 g 措施 0 <
lim 啜 t→0 <
∞
。
Let X be an Ahlfors d-regular space and mad-regular measure on X . We prove that a measure μ on X is d-homogeneous if and only if μ is mutually absolutely continuous with respect to m and the derivative Dmμ(x) is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.
