Book contents
- Frontmatter
- Contents
- Preface
- 1 Stringology
- 2 Number Theory and Algebra
- 3 Numeration Systems
- 4 Finite Automata and Other Models of Computation
- 5 Automatic Sequences
- 6 Uniform Morphisms and Automatic Sequences
- 7 Morphic Sequences
- 8 Frequency of Letters
- 9 Characteristic Words
- 10 Subwords
- 11 Cobham's Theorem
- 12 Formal Power Series
- 13 Automatic Real Numbers
- 14 Multidimensional Automatic Sequences
- 15 Automaticity
- 16 k-Regular Sequences
- 17 Physics
- Appendix Hints, References, and Solutions for Selected Exercises
- Bibliography
- Index
11 - Cobham's Theorem
Published online by Cambridge University Press: 13 October 2009
- Frontmatter
- Contents
- Preface
- 1 Stringology
- 2 Number Theory and Algebra
- 3 Numeration Systems
- 4 Finite Automata and Other Models of Computation
- 5 Automatic Sequences
- 6 Uniform Morphisms and Automatic Sequences
- 7 Morphic Sequences
- 8 Frequency of Letters
- 9 Characteristic Words
- 10 Subwords
- 11 Cobham's Theorem
- 12 Formal Power Series
- 13 Automatic Real Numbers
- 14 Multidimensional Automatic Sequences
- 15 Automaticity
- 16 k-Regular Sequences
- 17 Physics
- Appendix Hints, References, and Solutions for Selected Exercises
- Bibliography
- Index
Summary
As we have seen (Theorem 5.4.2), every ultimately periodic sequence is k-automatic for all integers k ≥ 2. In this chapter we prove a beautiful and deep theorem due to Cobham, which states that if a sequence s = (s(n))n≥0 is both k-automatic and l-automatic and k and l are multiplicatively independent, then S is ultimately periodic. (Recall that Theorem 2.5.7 discusses when two integers are multiplicatively independent.)
Syndetic and Right Dense Sets
In this section, we prove some useful preliminary results.
We say that a set X ⊆ Σ* is right dense if for any word u ∈ Σ* there exists a υ ∈ Σ* such that uυ ∈ X (that is, any word appears as a prefix of some word in X).
Lemma 11.1.1Let k, l ≥ 2 be multiplicatively independent integers, and let X be an infinite k-automatic set of integers. Then 0*(X)l = 0*{(n)l : n ∈ X} is right dense.
Proof. Since X is infinite and k-automatic, by the pumping lemma there exist strings t, u, υ with u nonempty such that tu*υ ⊆ (X)k. Let x ∈ {0, 1, …, l — 1}*. Our goal is to construct y such that xy ∊ 0*(X)l.
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- Information
- Automatic SequencesTheory, Applications, Generalizations, pp. 345 - 350Publisher: Cambridge University PressPrint publication year: 2003