For a real analytic periodic function \(\phi :\mathbb {R}\rightarrow \mathbb {R}\), an integer \(b\ge 2\) and \(\lambda \in (1/b,1)\), we prove the following dichotomy for the Weierstrass-type function \(W(x)=\sum \nolimits _{n\ge 0}{{\lambda }^n\phi (b^nx)}\): Either W(x) is real analytic, or the Hausdorff dimension of its graph is equal to \(2+\log _b\lambda \). Furthermore, given b and \(\phi \), the former alternative only happens for finitely many \(\lambda \) unless \(\phi \) is constant.

中文翻译：

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Weierstrass 型函数的二分法

对于实解析周期函数\(\phi :\mathbb {R}\rightarrow \mathbb {R}\)，整数\(b\ge 2\)和\(\lambda \in (1/b,1) \)，我们证明 Weierstrass 型函数\(W(x)=\sum \nolimits _{n\ge 0}{{\lambda }^n\phi (b^nx)}\)的以下二分法：要么w ^（X）是实解析，或它的图形的Hausdorff尺寸等于\（2+ \登录_b \拉姆达\） 。此外，给定b和\(\phi \)，除非\(\phi \)是常数，否则前一种选择只会发生有限多个\(\lambda \)。