文摘
I prove that if ∅≠K⊂R2∅≠K⊂R2 is a compact s xA0;-Ahlfors–David regular set with s≥1s≥1, thendimpD(K)=1,dimpD(K)=1, where D(K):={|x−y|:x,y∈K}D(K):={|x−y|:x,y∈K} is the distance set of K , and dimpdimp stands for packing dimension.The same proof strategy applies to other problems of similar nature. For instance, one can show that if ∅≠K⊂R2∅≠K⊂R2 is a compact s -Ahlfors–David regular set with s≥1s≥1, then there exists a point x0∈Kx0∈K such that dimpK⋅(K−x0)=1dimpK⋅(K−x0)=1.