14 research outputs found
Anti-Powers in Infinite Words
In combinatorics of words, a concatenation of consecutive equal blocks is
called a power of order . In this paper we take a different point of view
and define an anti-power of order as a concatenation of consecutive
pairwise distinct blocks of the same length. As a main result, we show that
every infinite word contains powers of any order or anti-powers of any order.
That is, the existence of powers or anti-powers is an unavoidable regularity.
Indeed, we prove a stronger result, which relates the density of anti-powers to
the existence of a factor that occurs with arbitrary exponent. As a
consequence, we show that in every aperiodic uniformly recurrent word,
anti-powers of every order begin at every position. We further show that every
infinite word avoiding anti-powers of order is ultimately periodic, while
there exist aperiodic words avoiding anti-powers of order . We also show
that there exist aperiodic recurrent words avoiding anti-powers of order .Comment: Revision submitted to Journal of Combinatorial Theory Series
Anti-Powers in Infinite Words
In combinatorics of words, a concatenation of k consecutive equal blocks is called a power of order k. In this paper we take a different point of view and define an anti-power of order k as a concatenation of k consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. From these results, we derive that at every position of an aperiodic uniformly recurrent word start anti-powers of any order. We further show that any infinite word avoiding anti-powers of order 3 is ultimately periodic, and that there exist aperiodic words avoiding anti-powers of order 4. We also show that there exist aperiodic recurrent words avoiding anti-powers of order 6, and leave open the question whether there exist aperiodic recurrent words avoiding anti-powers of order k for k=4,5
Anti-Power -fixes of the Thue-Morse Word
Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a
-anti-power, which is defined as a word of the form , where are distinct words of the
same length. For an infinite word and a positive integer , define
to be the set of all integers such that is a -anti-power, where denotes the -th letter of .
Define also ,
where denotes the Thue-Morse word. For all ,
is a well-defined positive integer,
and for sufficiently large, is a well-defined odd positive
integer. In his 2018 paper, Defant shows that and
grow linearly in . We generalize Defant's methods to prove that
and grow linearly in for any nonnegative
integer . In particular, we show that and . Additionally, we show
that and
.Comment: 19 page