A Note on the Abelian Complexity of the Rudin-Shapiro Sequence

Round 1

Reviewer 1 Report

This paper establishes the asymptotic growth of the abelian complexity
function of the Rudin-Shapiro sequence.  Two finite words are "abelian
equivalent" if one can be obtained by permuting the letters of the
other.  The "abelian complexity function" of an infinite word x counts
the numbers of abelian equivalence classes of the factors of x of
length n.  This is an important complexity measure for infinite
words.

The Rudin-Shapiro sequence r is a famous sequence defined by the
recurrence r(0)=1, r(2n)=r(n), r(2n+1)=(-1)^n*r(n).  It is an example
of a 2-automatic sequence, and its properties have been extensively
studied.  Lu, Chen, Wen, and Wu previously found a set of recurrence
relations that completely determine the abelian complexity function of
the Rudin-Shapiro sequence, but they did not determine the asymptotic
growth of this function.

In this paper the authors use the recurrence relations found by Lu, et
al. as a starting point to obtain the asymptotic behaviour of this
function.  In particular, they show that its values are between
\sqrt{3}\sqrt{n} and 3\sqrt{n}, and that these upper and lower bounds
are in fact optimal.  This is a nice note with a good result.  The
proofs are clear and correct.

I have noted a few minor typos/grammar mistakes:

- p. 2, line 13: and the author -> and the *first* author

- p. 2, line 20: of *the* Rudin-Shapiro

- p. 4, between line 44 and 45: Following from *the* inductive
  assumption

- p. 4, just after line 50: and *the* inductive assumption

- p. 6, a little before line 69: if follow from -> it follows from

- p. 8, after line 74, point 1 and point 2: this follows that -> and
  it follows that

- p. 9, line 87: Hence that we obtain -> Hence we obtain

Author Response

Thanks for your comments and suggestions. We have made corrections according to the
Reviewer’s comments as follows.
• “the author” is replaced by “the first author”
• “Rudin-Shapiro” is replaced by “the Rudin-Shapiro”
• “inductive assumption” is replaced by “the inductive assumption”
• “if follow from” is replaced by “it follows from”
• “this follow that” is replaced by “and it follows that”
• “Hence that we obtain” is replaced by “Hence we obatin”
Other typos are aslo fixed

Author Response File: Author Response.pdf

Reviewer 2 Report

This is a short paper examining the properties of a certain sequence of integers  defined in relation to the Rudin-Shapiro sequence.  A recurrence relation is derived for the sequence and as a result, many of the proofs are understandable and reasonable. 

The authors have motivated the work and have provided references for related papers. The work seems entirely reasonable and I have no suggestions for improvement.

Author Response

    We deeply appreciate the time and effort you have spent in reviewing our manuscript entitled  ``A note on the abelian complexity of the Rudin-Shapiro sequence'' (ID: Mathematics-1536885).

    Thank you for your positive comments and affirmation of our manuscript. In the revised version, a few minor typos and grammar mistakes are modified.

Author Response File: Author Response.pdf