397 research outputs found
Periodicity, repetitions, and orbits of an automatic sequence
We revisit a technique of S. Lehr on automata and use it to prove old and new
results in a simple way. We give a very simple proof of the 1986 theorem of
Honkala that it is decidable whether a given k-automatic sequence is ultimately
periodic. We prove that it is decidable whether a given k-automatic sequence is
overlap-free (or squareefree, or cubefree, etc.) We prove that the
lexicographically least sequence in the orbit closure of a k-automatic sequence
is k-automatic, and use this last result to show that several related
quantities, such as the critical exponent, irrationality measure, and
recurrence quotient for Sturmian words with slope alpha, have automatic
continued fraction expansions if alpha does.Comment: preliminary versio
Introduction: Special volume in honor of Jeffrey Shallit on the occasion of his 60th birthday
http://math.colgate.edu/~integers/ShallitIntro.pd
Decidability of the HD0L ultimate periodicity problem
In this paper we prove the decidability of the HD0L ultimate periodicity
problem
The Critical Exponent is Computable for Automatic Sequences
The critical exponent of an infinite word is defined to be the supremum of
the exponent of each of its factors. For k-automatic sequences, we show that
this critical exponent is always either a rational number or infinite, and its
value is computable. Our results also apply to variants of the critical
exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni
and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes
or recovers previous results of Krieger and others, and is applicable to other
situations; e.g., the computation of the optimal recurrence constant for a
linearly recurrent k-automatic sequence.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
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