发表机构
Bonn University Institute for Applied Mathematics（波恩大学应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究保持正规性的算术运算与映射，给出乘法情形的新证明并推广至加法情形，证明局部C^2保持映射必为仿射线性，并构造非仿射C^{1,1}反例。
AI 中文摘要
我们研究保持正规性的算术运算和映射。重新审视Dayan、Ganguly和Weiss最近的一个论证，我们给出了一个简化的、自包含的概率证明，说明在固定整数基下，非零有理数恰好是乘法保持正规性的数。我们还利用Hochman和Shmerkin的定理，将其扩展到所有基同时成立的情形。在加法情形下，我们将Rauzy关于固定整数基下确定性数的刻画推广到绝对正规数。即，一个平移保持绝对正规性当且仅当其参数在每个整数基下都是确定性的。我们通过一个稀疏随机扰动论证来证明这一推广。最后，我们证明所有局部$C^2$的正规性保持映射必须是仿射线性的。正则性假设不能减弱到$C^{1,1}$：我们构造了一个非仿射的$C^{1,1}$微分同胚，它在每个整数基下都保持正规性，因此保持绝对正规性。
英文摘要
We study arithmetic operations and maps which preserve normality. Revisiting a recent argument of Dayan, Ganguly and Weiss, we give a streamlined, self-contained presentation of the probabilistic proof that the non-zero rationals are exactly the multiplicative normality preserving numbers in a fixed integer base. We also include the extension to all bases simultaneously, using the theorem of Hochman and Shmerkin. In the additive case, we extend Rauzy's characterization of deterministic numbers in a fixed integer base to absolutely normal numbers. Namely, a translation preserves absolute normality if and only if its parameter is deterministic in every integer base. We prove this extension by a sparse random perturbation argument. Finally, we show that all locally $C^2$ normality preserving maps need to be affine-linear. The regularity assumption cannot be weakened to $C^{1,1}$: we construct a non-affine $C^{1,1}$ diffeomorphism which preserves normality in every integer base, and hence absolute normality.
Comments24 pages
