整数中分形集的加法与几何横截性
Additive and geometric transversality of fractal sets in the integers
AI总结：
本文在整数离散背景下定义对应于环面 $\times r$ 不变集的分形集，证明同时不变集分类、交集与和集横截性及迭代和集维数结果。
摘要：
通过并列分形几何与动力系统的思想，Furstenberg 在20世纪60年代末提出了一系列猜想，探究关于乘法独立基的数字展开之间的关系。在本工作中，我们在整数这一离散背景下引入并研究围绕 Furstenberg 工作的一些概念与结果的类比。特别地，我们定义了一类新的整数分形集，它对应于1维环面上的 $\times r$ 不变集概念，并考察当两个此类分形集分别依据乘法独立基构造时，它们之间的加法与几何独立性。我们在该方向上的主要结果对应于 Furstenberg、Hochman-Shmerkin、Shmerkin、Wu 以及 Lindenstrauss-Meiri-Peres 的工作，包括：对所有同时为 $\times r$ 不变与 $\times s$ 不变的正整数子集进行分类；给出 Furstenberg 两个横截性猜想的整数类比，这些猜想涉及当 $r$ 与 $s$ 乘法独立时，$\times r$ 不变集 $A$ 与 $\times s$ 不变集 $B$ 的交集 $A\cap B$ 以及和集 $A+B$ 的维数；以及对任意 $\times r$ 不变集 $A$，描述迭代和集 $A+A+\cdots+A$ 的维数。我们通过结合分形几何与遍历理论的思想，在连续情形与离散情形之间建立桥梁，从而获得这些结果。对于横截性结果，我们在很大程度上依赖 Shmerkin 最近得到的、关于受限数字 Cantor 测度投影的 $L^q$ 维数的定量界。最后，我们概述了若干关于整数分形子集的开放问题与研究方向。
英文摘要：
By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study - in the discrete context of the integers - analogues of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of $\times r$-invariant sets on the 1-torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman-Shmerkin, Shmerkin, Wu, and Lindenstrauss-Meiri-Peres and include: -a classification of all subsets of the positive integers that are simultaneously $\times r$- and $\times s$-invariant; -integer analogues of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection $A\cap B$ and the sumset $A+B$ of $\times r$- and $\times s$-invariant sets $A$ and $B$ when $r$ and $s$ are multiplicatively independent; and -a description of the dimension of iterated sumsets $A+A+\cdots+A$ for any $\times r$-invariant set $A$. We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the $L^q$-dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.
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