AI 中文总结
研究关于将有理数集映射到自身的\(C^k\)函数及马勒关于刘维尔数的问题。通过构造特定\(C^k\)函数集证明其保持刘维尔数，又证明了满足一定条件的\(C^{2k + 1}\)函数的刚性结果。
AI 中文摘要
刘维尔数是一类经典的超越实数，具有极强的有理逼近性。马利特定理表明有理系数的非恒定有理函数保持刘维尔性质，这引发了马勒关于超越函数是否有类似现象的问题。本文针对有限光滑度的实函数解决此问题。对于任意\(\varepsilon>0\)，构造了一族\(\mathbb{R}\)上的\(C^k\)函数，它们在紧集上一致收敛拓扑下稠密，将有理数集映射到自身且满足\(\text{den}(f(p/q)) \leq q^{2k+\varepsilon}\)，并由此推出这类函数保持刘维尔数。相反，证明了一个关于将有理数集映射到自身且满足\(\text{den}(f(p/q)) \ll q^k\)的\(C^{2k + 1}\)函数的刚性结果。
英文摘要
Liouville numbers form a classical class of transcendental real numbers characterized by exceptionally strong rational approximations. A theorem of Maillet shows that non-constant rational functions with rational coefficients preserve the Liouville property, motivating a question of Mahler on whether analogous phenomena hold for transcendental functions. In this paper, we address this problem for real functions of finite smoothness. For any $\varepsilon>0$, we construct an uncountable set of $C^k$-functions on $\mathbb{R}$, dense with respect to the topology of uniform convergence on compact sets, mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \le q^{2k+\varepsilon}$, and deduce that such functions preserve Liouville numbers. In contrast, we prove a rigidity result about a $C^{2k+1}$-function mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \ll q^k$.
Comments11 pages
