A Dichotomy for the Weierstrass-type functions

• 成果类型: Article
• 署名作者: Ren, Haojie; Shen, Weixiao
• 署名单位: Fudan University; Fudan University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01060-2
• 发表日期: 2021
• 页码: 1057-1100
• 关键词: hausdorff dimension; graphs; sets

摘要

For a real analytic periodic function phi : R -> R, an integer b >= 2 and lambda is an element of (1/b, 1), we prove the following dichotomy for the Weierstrass-type function W(x) = Sigma(n >=)0 lambda(n)phi(b(n)x): 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.