25 research outputs found
Enumeration and Decidable Properties of Automatic Sequences
We show that various aspects of k-automatic sequences -- such as having an
unbordered factor of length n -- are both decidable and effectively enumerable.
As a consequence it follows that many related sequences are either k-automatic
or k-regular. These include many sequences previously studied in the
literature, such as the recurrence function, the appearance function, and the
repetitivity index. We also give some new characterizations of the class of
k-regular sequences. Many results extend to other sequences defined in terms of
Pisot numeration systems
On the Number of Unbordered Factors
We illustrate a general technique for enumerating factors of k-automatic
sequences by proving a conjecture on the number f(n) of unbordered factors of
the Thue-Morse sequence. We show that f(n) = 4 and that f(n) = n
infinitely often. We also give examples of automatic sequences having exactly 2
unbordered factors of every length
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
The Simplest Binary Word with Only Three Squares
We re-examine previous constructions of infinite binary words containing few
distinct squares with the goal of finding the "simplest", in a certain sense.
We exhibit several new constructions. Rather than using tedious case-based
arguments to prove that the constructions have the desired property, we rely
instead on theorem-proving software for their correctness