On the distance sets of Ahlfors-David regular sets

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• 作者：Tuomas Orponen¹ ; ^{tuomas.orponen@helsinki.fi}
• 关键词：primary ; 28A80 ; secondary ; 28A78
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：307
• 期：Complete
• 页码：1029-1045
• 全文大小：379 K
• 卷排序：307

文摘

I prove that if ∅≠K⊂R2∅≠K⊂R2 is a compact s  -Ahlfors–David regular set with s≥1s≥1, thendimp⁡D(K)=1,dimp⁡D(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 dimp⁡K⋅(K−x0)=1dimp⁡K⋅(K−x0)=1.