2 research outputs found
Van der Waerden's Theorem and Avoidability in Words
Pirillo and Varricchio, and independently, Halbeisen and Hungerbuhler
considered the following problem, open since 1994: Does there exist an infinite
word w over a finite subset of Z such that w contains no two consecutive blocks
of the same length and sum? We consider some variations on this problem in the
light of van der Waerden's theorem on arithmetic progressions.Comment: Co-author added; new result
Avoiding Abelian powers in binary words with bounded Abelian complexity
The notion of Abelian complexity of infinite words was recently used by the
three last authors to investigate various Abelian properties of words. In
particular, using van der Waerden's theorem, they proved that if a word avoids
Abelian -powers for some integer , then its Abelian complexity is
unbounded. This suggests the following question: How frequently do Abelian
-powers occur in a word having bounded Abelian complexity? In particular,
does every uniformly recurrent word having bounded Abelian complexity begin in
an Abelian -power? While this is true for various classes of uniformly
recurrent words, including for example the class of all Sturmian words, in this
paper we show the existence of uniformly recurrent binary words, having bounded
Abelian complexity, which admit an infinite number of suffixes which do not
begin in an Abelian square. We also show that the shift orbit closure of any
infinite binary overlap-free word contains a word which avoids Abelian cubes in
the beginning. We also consider the effect of morphisms on Abelian complexity
and show that the morphic image of a word having bounded Abelian complexity has
bounded Abelian complexity. Finally, we give an open problem on avoidability of
Abelian squares in infinite binary words and show that it is equivalent to a
well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerb\"uhler.Comment: 16 pages, submitte