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Slices of the Takagi function

Part of: Classical measure theory Real functions

Published online by Cambridge University Press:  20 December 2023

ROOPE ANTTILA ,
BALÁZS BÁRÁNY  and
ANTTI KÄENMÄKI Open the ORCID record for ANTTI KÄENMÄKI [Opens in a new window]
ROOPE ANTTILA
Affiliation:
Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, FI-90014 Oulu, Finland (e-mail: roope.anttila@oulu.fi)
BALÁZS BÁRÁNY
Affiliation:
Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1111 Budapest, Hungary (e-mail: balubsheep@gmail.com)
ANTTI KÄENMÄKI*
Affiliation:
Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, FI-90014 Oulu, Finland
*
e-mail: antti.kaenmaki@oulu.fi
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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.

Keywords

Takagi functionself-affine setMarstrand’s slicing theoremAssouad dimension

MSC classification

Primary: 28A80: Fractals
Secondary: 28A75: Length, area, volume, other geometric measure theory 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 9 , September 2024 , pp. 2361 - 2398
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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