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Gap sequences of McMullen sets


Authors: Jun Jie Miao, Li-Feng Xi and Ying Xiong
Journal: Proc. Amer. Math. Soc. 145 (2017), 1629-1637
MSC (2010): Primary 28A80
DOI: https://doi.org/10.1090/proc/13342
Published electronically: November 21, 2016
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Abstract: We study the gap sequence of totally disconnected McMullen sets. Our result shows that if every horizontal line in the McMullen set is nonempty, then the gap sequence is unrelated to the box dimension. This implies that in such situations, the separation properties of McMullen sets are quite different from that of self-similar sets.


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Jun Jie Miao
Affiliation: Shanghai Key Laboratory of PMMP, Department of Mathematics, East China Normal University, No. 500, Dongchuan Road, Shanghai 200241, People’s Republic of China
Email: jjmiao@math.ecnu.edu.cn

Li-Feng Xi
Affiliation: Department of Mathematics, Ningbo University, Ningbo 315211, People’s Republic of China
Email: xilifengningbo@yahoo.com

Ying Xiong
Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China
Email: xiongyng@gmail.com

DOI: https://doi.org/10.1090/proc/13342
Keywords: McMullen sets, gap sequences
Received by editor(s): June 7, 2015
Received by editor(s) in revised form: June 5, 2016
Published electronically: November 21, 2016
Additional Notes: The authors were supported by National Natural Science Foundation of China (Grants No. 11201152, 11371329, 11471124), NSF of Zhejiang Province (No. LR13A010001), the Fund for the Doctoral Program of Higher Education of China 20120076120001 and Morningside Center of Mathematics.
Communicated by: Kevin Whyte
Article copyright: © Copyright 2016 American Mathematical Society