Slices of the Takagi function
Anttila, Roope; Barany, Balazs; Käenmäki, Antti (2023-12-20)
Cambridge University Press
20.12.2023
ANTTILA, R., BÁRÁNY, B., & KÄENMÄKI, A. (2024). Slices of the Takagi function. Ergodic Theory and Dynamical Systems, 44(9), 2361–2398. doi:10.1017/etds.2023.117
https://creativecommons.org/licenses/by-nc-nd/4.0/
This article has been published in a revised form in Ergodic theory and dynamical systems https://doi.org/10.1017/etds.2023.117. This version is published under a Creative Commons CC-BY-NC-ND licence. No commercial re-distribution or re-use allowed. Derivative works cannot be distributed. © The Author(s), 2023. Published by Cambridge University Press.
https://creativecommons.org/licenses/by-nc-nd/4.0/
This article has been published in a revised form in Ergodic theory and dynamical systems https://doi.org/10.1017/etds.2023.117. This version is published under a Creative Commons CC-BY-NC-ND licence. No commercial re-distribution or re-use allowed. Derivative works cannot be distributed. © The Author(s), 2023. Published by Cambridge University Press.
https://creativecommons.org/licenses/by-nc-nd/4.0/
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202406204832
https://urn.fi/URN:NBN:fi:oulu-202406204832
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Abstract
We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.
We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.
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