Slices of the Takagi function

Roope Anttila; Balázs Bárány; Antti Käenmäki

doi:10.1017/etds.2023.117

Slices of the Takagi function

Roope Anttila, Balázs Bárány, Antti Käenmäki^*

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	2361-2398
Number of pages	38
Journal	Ergodic Theory and Dynamical Systems
Volume	44
Issue number	9
Early online date	20 Dec 2023
DOIs	https://doi.org/10.1017/etds.2023.117
Publication status	Published - 1 Sept 2024

Keywords

Takagi function
Self-affine set
Marstrand's slicing theorem
Assouad dimension

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10.1017/etds.2023.117

https://arxiv.org/abs/2305.08181v2

Cite this

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title = "Slices of the Takagi function",

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{\textquoteright}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.",

keywords = "Takagi function, Self-affine set, Marstrand's slicing theorem, Assouad dimension",

author = "Roope Anttila and Bal{\'a}zs B{\'a}r{\'a}ny and Antti K{\"a}enm{\"a}ki",

note = "Funding: R. Anttila was financially supported by the Magnus Ehrnrooth foundation. B. B{\'a}r{\'a}ny was financially supported by the grants NKFI FK134251, K142169, and the grant NKFI KKP144059 {\textquoteleft}Fractal geometry and applications{\textquoteright}.",

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N2 - 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.

AB - 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.

KW - Takagi function

KW - Self-affine set

KW - Marstrand's slicing theorem

KW - Assouad dimension

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SP - 2361

EP - 2398

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 9

ER -

Slices of the Takagi function

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Keywords

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