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Tyson型集及Borel函数的图的拟对称极小性
Quasisymmetric Minimality of Sets of Tyson Type and Graphs of Borel Functions
本文将欧氏空间Rd中形如[0，1]×Z的集称为Tyson型集，其中d>1，Z⊂Rd-1.已知当Z是Rd-1中的紧集时，Tyson型集是拟对称极小集.本文改进了这个结果，证明了当Z是Rd-1中的Borel集时，Tyson型集仍是拟对称极小集.作为应用，我们证明了Tyson型集三个形变版本的拟对称极小性，其中一个结果是：设Z是Rd-1中的任一Borel集，h：Z→R1是Borel函数，满足dimH（{h≠0}∩Z）=dimH Z，则h的图G（h）是拟对称极小集，其中h的图G（h）定义为G（h）={（z，y）：z∈Z，y∈[0，h（z）]}.
In this paper, sets of the form[0,1]×Z in Rd will be called sets of Tyson type, where d > 1 and Z ⊂ Rd-1. It is known that every set E of Tyson type is quasisymmetrically minimal, provided that Z is compact in Rd-1. We prove that this result holds by only assuming that Z is Borel in Rd-1. As applications, we prove that the quasisymmetric minimality of three variants of sets of Tyson type. One of the results is that the graph G(h) of h is quasisymmetrically minimal, provided that Z is a Borel set in Rd-1 and h:Z → R1 is a Borel function satisfying the condition dimH({h ≠ 0} ∩ Z)=dimH Z, where G(h)={(z, y):z ∈ Z, y ∈[0, h(z)]}.
Tyson型集 / Hausdorff维数 / 拟对称极小集 {{custom_keyword}} /
sets of Tyson type / Hausdorff dimension / quasisymmetrically minimal sets {{custom_keyword}} /
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国家自然科学基金资助项目（11871200，11671189）；山西省高等学校科技创新项目（2019L0963）
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