论文标题
Weierstrass型功能的二分法
A Dichotomy for the Weierstrass-type functions
论文作者
论文摘要
对于真实的分析周期性函数$ ϕ：\ Mathbb {r} \ to \ Mathbb {r} $，一个整数$ b \ ge 2 $和$λ\ in（1/b，1）$，我们证明了以下WeierStrass-Type函数$ W（x）= \ sum \ sum \ sum \ sum \ libirs_ 0} {λ^n ϕ（b^nx）} $：$ w（x）$是真实的分析性，或者其图的Hausdorff尺寸等于$ 2+\log_bλ$。此外，给定$ b $和$ ϕ $，除非$ ϕ $是恒定的，否则以前的替代方案仅对有限的$λ$进行。
For a real analytic periodic function $ϕ:\mathbb{R}\to \mathbb{R}$, an integer $b\ge 2$ and $λ\in (1/b,1)$, we prove the following dichotomy for the Weierstrass-type function $W(x)=\sum\limits_{n\ge 0}{λ^nϕ(b^nx)}$: Either $W(x)$ is real analytic, or the Hausdorff dimension of its graph is equal to $2+\log_bλ$. Furthermore, given $b$ and $ϕ$, the former alternative only happens for finitely many $λ$ unless $ϕ$ is constant.
