Avoiding large squares in infinite binary words

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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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Cubefree binary words avoiding long squares

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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Avoiding large squares in partial words

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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On infinite words avoiding a finite set of squares

Building an infinite square-free word by appending one letter at a time while simultaneously avoiding the creation of squares is most likely to fail. When the alphabet has two letters this approach is impossible. When the alphabet has three or more letters, one will most probably create a word in which the addition of any letter invariably creates a square. When one restricts the set of undesir...

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ژورنال

عنوان ژورنال: Theoretical Computer Science

سال: 2005

ISSN: 0304-3975

DOI: 10.1016/j.tcs.2005.01.005