AI 中文总结
该研究证明局部有限Borel测度的共形维数非零即无穷，提出拟对称降维定理，还构造了满足特定Hausdorff维数约束的度量空间与Borel集，得到相关界的最优性结论。
AI 中文摘要
我们证明了每个局部有限Borel测度的共形维数要么为0要么为无穷大。核心工具是一个拟对称降维定理：可分度量空间上每个具有有限Hausdorff维数的全支撑概率测度，都存在拟对称等价的度量，使得其Hausdorff维数可任意小。特别地，加倍度量空间上的每个局部有限Borel测度的共形维数都为0。我们还证明，对每个n≥1和p>0，存在度量空间X、拟对称同胚f:[0,1]^n→X及Borel集E⊂[0,1]^n，使得dim_H f(E)≤p且dim_H([0,1]^n\backslash E)≤n-1+p；第二个界关于p是最优的：若dim_H f(E)<1，则dim_H([0,1]^n\backslash E)≥n-1。
英文摘要
We prove that the conformal dimension of every locally finite Borel measure is either zero or infinite. The main ingredient is a quasisymmetric dimension-reduction theorem: every full-support probability measure of finite Hausdorff dimension on a separable metric space admits quasisymmetrically equivalent metrics in which its Hausdorff dimension is arbitrarily small. In particular, every locally finite Borel measure on a doubling metric space has conformal dimension zero. We also prove that for every $n\geq 1$ and $p>0$ there are a metric space $X$, a quasisymmetric homeomorphism $f:[0,1]^n\to X$, and a Borel set $E\subset[0,1]^n$ such that $\dim_H f(E)\leq p$ and $\dim_H([0,1]^n\setminus E)\leq n-1+p$. The second bound is sharp up to $p$: if $\dim_H f(E)<1$, then $\dim_H([0,1]^n\setminus E)\geq n-1$.
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