A generalization of the Takagi function for beta-expansions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917420878266368 |
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| author | Suzuki, Shintaro |
| author_facet | Suzuki, Shintaro |
| contents | We consider a generalized Takagi function for beta-expansions with the base $1<β\leq2$, motivated by multifractal analysis for digit frequency sets of beta-expansions [20]. We show that it is pointwise $α$-Hölder continuous for any $α\in(0,1)$ but not pointwise Lipschitz continuous on the unit interval except a Lebesgue null set. Our proof relies on a formula for the generalized Takagi function reflecting its oscillations of the sum of digits and some basic limit theorems for the corresponding beta-map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17809 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A generalization of the Takagi function for beta-expansions Suzuki, Shintaro Dynamical Systems We consider a generalized Takagi function for beta-expansions with the base $1<β\leq2$, motivated by multifractal analysis for digit frequency sets of beta-expansions [20]. We show that it is pointwise $α$-Hölder continuous for any $α\in(0,1)$ but not pointwise Lipschitz continuous on the unit interval except a Lebesgue null set. Our proof relies on a formula for the generalized Takagi function reflecting its oscillations of the sum of digits and some basic limit theorems for the corresponding beta-map. |
| title | A generalization of the Takagi function for beta-expansions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2604.17809 |