Avoiding large squares in infinite binary words
نویسندگان
چکیده
منابع مشابه
Avoiding large squares in infinite binary words
We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with |y| ≥ 4; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary word avoiding all squares except 0, 1, and (01); our construction is somewhat simpler than the origina...
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Entringer, Jackson, and Schatz conjectured in 1974 that every infinite cubefree binary word contains arbitrarily long squares. In this paper we show this conjecture is false: there exist infinite cubefree binary words avoiding all squares xx with |x| ≥ 4, and the number 4 is best possible. However, the Entringer-Jackson-Schatz conjecture is true if “cubefree” is replaced with “overlap-free”.
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Well-known results on the avoidance of large squares in (full) words include the following: (1) Fraenkel and Simpson showed that we can construct an infinite binary word containing at most three distinct squares; (2) Entringer, Jackson and Schatz showed that there exists an infinite binary word avoiding all squares of the form xx such that |x| ≥ 3, and that the bound 3 is optimal; (3) Dekking s...
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In 1976, Dekking showed that there exists an infinite binary word that contains neither squares yy with |y| ≥ 4 nor cubes xxx. We show that ‘cube’ can be replaced by any fractional power > 5/2. We also consider the analogous problem where ‘4’ is replaced by any integer. This results in an interesting and subtle hierarchy.
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ژورنال
عنوان ژورنال: Theoretical Computer Science
سال: 2005
ISSN: 0304-3975
DOI: 10.1016/j.tcs.2005.01.005