AI 中文总结
本文研究魏尔斯特拉斯函数图像的Assouad维数公开问题，证明其严格小于2并给出定量上界，结论可推广至含Takagi函数的广义魏尔斯特拉斯函数族，同时否定了H. Yu的相关猜想。
AI 中文摘要
魏尔斯特拉斯函数族$W_{a,b}$是最早的一类连续且处处不可微的实函数例子之一。确定这类函数的图像$G(W_{a,b})$的各种维数是一项颇具挑战性的研究方向。例如，在众多学者取得一系列阶段性成果之后，Shen才于2018年最终确定了$G(W_{a,b})$的Hausdorff维数。尽管$G(W_{a,b})$的Assouad维数仍是J. M. Fraser提出的一个公开问题，但已有诸多迹象表明其值可能等于2。这些迹象包括维纳过程的图像、Baire范畴意义下几乎所有Hölder函数的图像，以及经过可数多次反射后的魏尔斯特拉斯函数图像，它们的Assouad维数均为2。本文证明事实并非如此，给出了$G(W_{a,b})$的Assouad维数的一个严格小于2的定量上界。特别地，我们证明该上界对一类广义魏尔斯特拉斯函数$W_{a,b}^ϕ(x) = \sum_{j=0}^\infty a^jϕ(b^j x)$成立，这类函数包含$W_{a,b}$以及Takagi函数族。后一结论还被用于否定回答H. Yu提出的一个关于参数$a\in (0,1)$、$b\in (1/a, \infty)\cap \mathbb{Z}$时Takagi函数图像Assouad维数的猜想。
英文摘要
The class of Weierstrass functions $W_{a,b}$ is one of the first class of examples of continuous and nowhere differentiable real functions. A challenging line of research has been to determine the various dimensions of graphs $G(W_{a,b})$ of such functions. For instance, the Hausdorff dimension of $G(W_{a,b})$ was only recently determined by Shen in 2018, after a long series of partial results by many different authors. While the Assouad dimension of $G(W_{a,b})$ remains an open problem, also posed as a question by J. M. Fraser, there have been many indications that it might be equal to $2$. Such indications include the graph of Wiener processes, the graphs of almost all Hölder functions in the Baire category sense, and the graphs of Weierstrass functions after a series of countably many reflections all having Assouad dimension equal to $2$. In this paper we show that this is not the case, providing a quantitative upper bound on the Assouad dimension of $G(W_{a,b})$ that is strictly less than $2$. In particular, we show that such a bound is true for a class of generalized Weierstrass functions $W_{a,b}^ϕ(x) = \sum_{j=0}^\infty a^jϕ(b^j x)$, which includes $W_{a,b}$ and the class of Takagi functions. The latter fact is used to also answer in the negative a conjecture of H. Yu on the Assouad dimension of graphs of Takagi functions for parameters $a\in (0,1)$, $b\in (1/a, \infty)\cap \mathbb{Z}$.
Comments26 pages, 1 figure
