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A Dichotomy for the Weierstrass-type functions


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.

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We would like to thank the participants of the dynamical systems seminar in the Shanghai Center for Mathematical Sciences, an in particular, Guohua Zhang for suggesting the name of regulating period. WS is supported by NSFC grant No. 11731003.

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Correspondence to Weixiao Shen.

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Ren, H., Shen, W. A Dichotomy for the Weierstrass-type functions. Invent. math. (2021).

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