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----
-template: oldpost
-title: AMS review of 'Infinite binary words containing repetitions of odd period' by Badkobeh and Crochemore
-date: 2015-05-30
-comments: true
-archive: false
----
-This paper is about the existence of pattern-avoiding infinite binary
-words, where the patterns are squares, cubes and \\(3^+\\)-powers.
-   There are mainly two kinds of results, positive (existence of an
-infinite binary word avoiding a certain pattern) and negative
-(non-existence of such a word). Each positive result is proved by the
-construction of a word with finitely many squares and cubes which are
-listed explicitly. First a synchronising (also known as comma-free)
-uniform morphism \\(g\\: \\Sigma_3^\* \\to
-\\Sigma_2^\*\\)
-
-is constructed. Then an argument is given to show that the length of
-squares in the code \\(g(w)\\) 
-for a squarefree \\(w\\) is bounded, hence all the squares can be obtained by examining all \\(g(s)\\) for \\(s\\)
-of bounded lengths. The argument resembles that of the proof of, e.g.,
-Theorem 1, Lemma 2, Theorem 3 and Lemma 4 in \[N. Rampersad, J. O.
-Shallit and M. Wang, Theoret. Comput. Sci. **339**
-(2005), no. 1, 19–34;
-[MR2142071](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=2142071&loc=fromrevtext)\].
-The negative results are proved by traversing all possible finite words
-satisfying the conditions.
-
-   Let \\(L(n_2, n_3, S)\\) be the maximum length of a word with \\(n_2\\) distinct squares, \\(n_3\\) 
-distinct cubes and that the periods of the squares can take values only
-in \\(S\\) 
-, where \\(n_2, n_3 \\in \\Bbb N \\cup
-\\{\\infty, \\omega\\}\\) 
-and \\(S \\subset \\Bbb N_+\\) 
-.
-   \\(n_k = 0\\) 
-corresponds to \\(k\\)-free, \\(n_k = \\infty\\) 
-means no restriction on the number of distinct \\(k\\)-powers, and \\(n_k = \\omega\\) 
-means \\(k^+\\)-free.
-
-   Below is the summary of the positive and negative results:
-
-(1) (Negative) \\(L(\\infty, \\omega, 2 \\Bbb N)
-    < \\infty\\)
-    : \\(\\nexists\\)
-    an infinite \\(3^+\\)
-    -free binary word avoiding all squares of odd periods.
-    (Proposition 1)
-
-(2) (Negative) \\(L(\\infty, 0, 2 \\Bbb N + 1) \\le
-    23\\)
-    : \\(\\nexists\\)
-    an infinite 3-free binary word, avoiding squares of even periods.
-    The longest one has length \\(\\le 23\\)
-    (Proposition 2).
-
-(3) (Positive) \\(L(\\infty, \\omega, 2 \\Bbb N +
-    1) = \\infty\\)
-    : \\(\\exists\\)
-    an infinite \\(3^+\\)
-    -free binary word avoiding squares of even periods (Theorem 1).
-
-(4) (Positive) \\(L(\\infty, \\omega, \\{1, 3\\}) =
-    \\infty\\)
-    : \\(\\exists\\)
-    an infinite \\(3^+\\)
-    -free binary word containing only squares of period 1 or 3
-    (Theorem 2).
-
-(5) (Negative) \\(L(6, 1, 2 \\Bbb N + 1) =
-    57\\)
-    : \\(\\nexists\\)
-    an infinite binary word avoiding squares of even period containing
-    \\(< 7\\)
-    squares and \\(< 2\\)
-    cubes. The longest one containing 6 squares and 1 cube has length 57
-    (Proposition 6).
-
-(6) (Positive) \\(L(7, 1, 2 \\Bbb N + 1) =
-    \\infty\\)
-    : \\(\\exists\\)
-    an infinite \\(3^+\\)
-    -free binary word avoiding squares of even period with 1 cube and 7
-    squares (Theorem 3).
-
-(7) (Positive) \\(L(4, 2, 2 \\Bbb N + 1) =
-    \\infty\\)
-    : \\(\\exists\\)
-    an infinite \\(3^+\\)
-    -free binary words avoiding squares of even period and containing 2
-    cubes and 4 squares (Theorem 4).
-
-
-Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3313467, its copyright owned by the AMS.