Two notes on subshifts

Author:
Joseph S. Miller

Journal:
Proc. Amer. Math. Soc. **140** (2012), 1617-1622

MSC (2010):
Primary 37B10, 03D30; Secondary 03D32, 68Q30

DOI:
https://doi.org/10.1090/S0002-9939-2011-11000-1

Published electronically:
August 31, 2011

MathSciNet review:
2869146

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove two unrelated results about subshifts. First, we give a condition on the lengths of forbidden words that is sufficient to guarantee that the corresponding subshift is nonempty. The condition implies that, for example, any sequence of binary words of lengths is avoidable. As another application, we derive a result of Durand, Levin and Shen that there are infinite sequences such that every substring has high Kolmogorov complexity. In particular, for any , there is a and an infinite binary sequence such that if is a substring of , then has Kolmogorov complexity greater than .

The second result says that from the standpoint of computability theory, any behavior possible from an arbitrary effectively closed subset of (i.e., a class) is exhibited by an effectively closed subshift. In technical terms, every Medvedev degree contains a subshift. This answers a question of Simpson.

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Additional Information

**Joseph S. Miller**

Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388

Email:
jmiller@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-11000-1

Received by editor(s):
March 17, 2010

Received by editor(s) in revised form:
August 12, 2010, and January 8, 2011

Published electronically:
August 31, 2011

Additional Notes:
The author was supported by the National Science Foundation under grants DMS-0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness.

Communicated by:
Julia Knight

Article copyright:
© Copyright 2011
American Mathematical Society

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