AI 中文总结
研究康托集交集维数下降问题，通过证明特定条件下\(f(E)\cap E\)的上闵可夫斯基维数小于\(E\)的豪斯多夫维数等，建立定量结果并刻画\(\gamma\)，为相关维数研究提供了新结论。
AI 中文摘要
设\(E\subset \mathbb{R}\)是由具有收缩率\(\rho\in (0,1)\)的齐次迭代函数系统\(\Phi\)生成的自相似集。假设\(\Phi\)满足开集条件且\(\dim_{\rm H}E<1\)。设\(f\)是\(\mathbb{R}\)上的\(C^1\) - 微分同胚。证明了若对于每个\(x\in E\cap f^{-1}(E)\)，\(\log|f'(x)|/\log\rho\not\in\mathbb{Q}\)，则\(f(E)\cap E\)的上闵可夫斯基维数严格小于\(E\)的豪斯多夫维数。还建立了\(E\)是缺数字集且\(f\)是满足特定算术条件的有理斜率仿射映射时的定量维数下降结果。基于这些结果和Shmerkin的一个结果，得到了在各种情况下\(\gamma\)的刻画，使得对于每个\(\alpha\in\mathbb{R}\)，\(\overline{\dim}_{\rm M}\left((\gamma E+\alpha)\cap E\right)<\dim_{\rm H}E\)。
英文摘要
Let $E\subset \mathbb{R}$ be a self-similar set generated by a homogeneous iterated function system $Φ$ with contraction ratio $ρ\in (0,1)$. Assume that $Φ$ satisfies the open set condition and $\dim_{\rm H}E<1$. Let $f$ be a $C^1$-diffeomorphism on $\mathbb{R}$. We prove that if $\log|f'(x)|/\logρ\not\in\mathbb{Q}$ for every $x\in E\cap f^{-1}(E)$, then the upper Minkowski dimension of $f(E)\cap E$ is strictly less than the Hausdorff dimension of $E$. We also establish a quantitative dimension drop result when $E$ is a missing-digit set and $f$ is an affine map with rational slope satisfying a certain arithmetic condition. Based on these results and a result of Shmerkin [Ann. of Math., 2019], we obtain characterizations of $γ$ in various contexts such that $\overline{\dim}_{\rm M}\left((γE+α)\cap E\right)<\dim_{\rm H}E$ for every $α\in\mathbb{R}$.
