Dimensions of the popcorn graph

Haipeng Chen; Jonathan Fraser; Han Yu

doi:10.1090/proc/15729

Dimensions of the popcorn graph

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

Abstract

The 'popcorn function' is a well-known and important example in real analysis with many interesting features. We prove that the box dimension of the graph of the popcorn function is 4/3, as well as computing the Assouad dimension and Assouad spectrum. The main ingredients include Duffin-Schaeffer type estimates from Diophantine approximation and the Chung-Erdős inequality from probability theory.

Original language	English
Pages (from-to)	4729-4742
Journal	Proceedings of the American Mathematical Society
Volume	150
Early online date	18 Aug 2022
DOIs	https://doi.org/10.1090/proc/15729
Publication status	E-pub ahead of print - 18 Aug 2022

Keywords

Popcorn function
Thomae function
Box dimension
Assouad dimension
Assouad spectrum

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10.1090/proc/15729

dimension_popcorn_final_arxiv
Copyright © 2021 American Mathematical Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://www.ams.org/cgi-bin/mstrack/accepted_papers/proc.
Accepted author manuscript, 420 KBLicence: CC BY-NC-ND

https://arxiv.org/abs/2007.08407

Cite this

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title = "Dimensions of the popcorn graph",

abstract = "The 'popcorn function' is a well-known and important example in real analysis with many interesting features. We prove that the box dimension of the graph of the popcorn function is 4/3, as well as computing the Assouad dimension and Assouad spectrum. The main ingredients include Duffin-Schaeffer type estimates from Diophantine approximation and the Chung-Erd{\H o}s inequality from probability theory.",

keywords = "Popcorn function, Thomae function, Box dimension, Assouad dimension, Assouad spectrum",

author = "Haipeng Chen and Jonathan Fraser and Han Yu",

note = "Funding: H. Chen is thankful for the excellent atmosphere for research provided by the University of St Andrews. H. Chen was funded by China Scholarship Council (File No. 201906150102) and NSFC (No. 11601161, 11771153 and 11871227). J. M. Fraser was supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034). H. Yu was supported by the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme (grant agreement No. 803711), and indirectly by Corpus Christi College, Cambridge. ",

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KW - Popcorn function

KW - Thomae function

KW - Box dimension

KW - Assouad dimension

KW - Assouad spectrum

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EP - 4742

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

ER -

Dimensions of the popcorn graph

Abstract

Keywords

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New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

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Dimensions of the popcorn graph

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this