A multi-dimensional analogue of Cobham's theorem for fractals

Authors:
Davy Ho-Yuen Chan and Kevin G. Hare

Journal:
Proc. Amer. Math. Soc. **142** (2014), 449-456

MSC (2010):
Primary 11B85, 28A80

DOI:
https://doi.org/10.1090/S0002-9939-2013-11843-5

Published electronically:
November 14, 2013

MathSciNet review:
3133987

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the set of hyper-cubes of size in with vertices having coordinates of the form . For we define as the linear expansion map from to in the obvious way. We extend the map to a map on all of by defining for .

Let be a compact subset of and be an integer. We define the *-kernel* of as

*finite -kernel*or, equivalently, that is

*-self-similar*. Some examples of this are the standard Cantor set, the Sierpiński carpet and the Sierpiński triangle.

Recently Adamczewski and Bell showed an analogue of Cobham's theorem for one-dimensional fractals. Let and be multiplicatively independent positive integers. They proved that the compact set is both - and -self-similar if and only if is a union of a finite number of intervals with rational endpoints.

In their paper, Adamczewski and Bell conjectured that a similar result should be true in higher dimensions. We prove their conjecture; in particular we prove:

**Theorem.** Let and be multiplicatively independent positive integers. The compact set is both - and -self-similar if and only if is a union of a finite number of rational polyhedra.

**[1]**Boris Adamczewski and Jason Bell,*An analogue of Cobham’s theorem for fractals*, Trans. Amer. Math. Soc.**363**(2011), no. 8, 4421–4442. MR**2792994**, https://doi.org/10.1090/S0002-9947-2011-05357-2**[2]**A. D. Alexandrov,*Convex polyhedra*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky; With comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov. MR**2127379****[3]**Jean-Paul Allouche and Jeffrey Shallit,*Automatic sequences*, Cambridge University Press, Cambridge, 2003. Theory, applications, generalizations. MR**1997038****[4]**Alan Cobham,*On the base-dependence of sets of numbers recognizable by finite automata*, Math. Systems Theory**3**(1969), 186–192. MR**0250789**, https://doi.org/10.1007/BF01746527**[5]**Kenneth Falconer,*Fractal geometry*, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR**2118797****[6]**A. L. Semenov,*The Presburger nature of predicates that are regular in two number systems*, Sibirsk. Mat. Ž.**18**(1977), no. 2, 403–418, 479 (Russian). MR**0450050**

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Additional Information

**Davy Ho-Yuen Chan**

Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Email:
chydavy@gmail.com

**Kevin G. Hare**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Email:
kghare@uwaterloo.ca

DOI:
https://doi.org/10.1090/S0002-9939-2013-11843-5

Received by editor(s):
November 16, 2011

Received by editor(s) in revised form:
March 28, 2012

Published electronically:
November 14, 2013

Additional Notes:
The research of the first author was supported by the Department of Mathematics, The Chinese University of Hong Kong

The research of the second author was partially supported by NSERC

Communicated by:
Ken Ono

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.