A Dichotomy for the Weierstrass-type functions

Haojie Ren School of Mathematical Sciences, Fudan University, No 220 Handan Road, Shanghai 200433, China Weixiao Shen Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan University, No 2005 Songhu Road, Shanghai 200438, China

Classical Analysis and ODEs mathscidoc:2203.05002

Inventiones mathematicae, 226, 1057-1100, 2021.7
For a real analytic periodic function 𝜙: ℝ→ℝ, an integer 𝑏≥2 and 𝜆∈(1/𝑏,1), we prove the following dichotomy for the Weierstrass-type function 𝑊(𝑥)=∑_{𝑛≥0} 𝜆^𝑛 𝜙(𝑏^𝑛 𝑥): Either W(x) is real analytic, or the Hausdorff dimension of its graph is equal to 2+log𝑏𝜆. Furthermore, given b and 𝜙, the former alternative only happens for finitely many 𝜆 unless 𝜙 is constant.
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  title={A Dichotomy for the Weierstrass-type functions},
  author={Haojie Ren, and Weixiao Shen},
  booktitle={Inventiones mathematicae},
Haojie Ren, and Weixiao Shen. A Dichotomy for the Weierstrass-type functions. 2021. Vol. 226. In Inventiones mathematicae. pp.1057-1100. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316102852746058971.
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