基于高斯类数的自相似集的几何分类

正如Falconer所述：利用双Lipschitz映射进行分形的几何分类，在分形几何学研究中居于重要地位。而满足开集条件的自相似集，是最经典的分形。尽管对开集条件的研究日益地深入，但因开集本身的复杂性，该条件的各种刻画仍不尽透彻。本研究尝试采用代数数论中的高斯"类数"理论、代数学的"模表示论"、处理开集条件的"block"新技术、"测度线性"准则，以及无理代数数的新的组合分拆技术，力图通过双Lipschitz几何等价分类的角度重新审视自相似集，并还原开集条件的几何面貌。本项研究将综合数论、代数学以及组合学等各种技巧，探索从双Lipschitz等价的新角度深入分形的几何研究，不仅揭示自相似集的几何结构的刚性定理，同时也为高斯"类数猜想"从几何角度进行研究提供了某种可能性。

As mentioned by Falconer, geometric classification by Lipschitz equivalence plays an important role in fractal geometry. The self-similar sets satisfying the open set condition (OSC) are the most remarkable fractals, where the OSC has not comprehended completely. Using Gauss's class number, modular representation theory, block technique for the OSC, measure linear criterion and new partition method of algebraic number, we revisit the self-similar sets and try to illuminate the intrinsic property of the OSC in the new viewpoint of geometric classification. Applying the techniques in number they, algebra and combinatorics, we not only obtain the new perspective to explore the rigidity theorems on geometric structure of self-similar sets, also provide a new possible geometric approach for Gauss's class number problems.