Dimensions of the popcorn graph

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.

Citation

Chen , H , Fraser , J & Yu , H 2022 , ' Dimensions of the popcorn graph ' , Proceedings of the American Mathematical Society , vol. 150 , pp. 4729-4742 . https://doi.org/10.1090/proc/15729

Publication

Proceedings of the American Mathematical Society

Status

Peer reviewed

DOI

https://doi.org/10.1090/proc/15729

ISSN

0002-9939

Type

Journal article

Rights

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.

Description

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’s Horizon 2020 research and innovation programme (grant agreement No. 803711), and indirectly by Corpus Christi College, Cambridge.