Anti-Power Prefixes of the Thue-Morse Word

نویسنده

  • Colin Defant
چکیده

Recently, Fici, Restivo, Silva, and Zamboni defined a k-anti-power to be a word of the form w1w2 · · ·wk, where w1, w2, . . . , wk are distinct words of the same length. They defined AP (x, k) to be the set of all positive integers m such that the prefix of length km of the word x is a k-anti-power. Let t denote the Thue-Morse word, and let F(k) = AP (t, k) ∩ (2Z+ − 1). For k > 3, γ(k) = min(F(k)) and Γ(k) = max((2Z+ − 1) \ F(k)) are well-defined odd positive integers. Fici et al. speculated that γ(k) grows linearly in k. We prove that this is indeed the case by showing that 1/2 6 lim inf k→∞ (γ(k)/k) 6 9/10 and 1 6 lim sup k→∞ (γ(k)/k) 6 3/2. In addition, we prove that lim inf k→∞ (Γ(k)/k) = 3/2 and lim sup k→∞ (Γ(k)/k) = 3.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Some Properties of Abelian Return Words

We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with nonzero intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue–Morse word. We also investigate t...

متن کامل

Subword complexity and power avoidance

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that – the Thue-Morse word has the minimum possible subword complexity over all overlapfree binary words and all (73)-power-free binary words, but not over all ( 7 3) +-power-free binary words; – the twisted Thue-Morse word has the maximum possible sub...

متن کامل

-powers and the Thue-Morse morphism

In 1982, Séébold showed that the only overlap-free binary words that are the fixed points of non-identity morphisms are the Thue-Morse word and its complement. We strengthen Séébold’s result by showing that the same result holds if the term ‘overlapfree’ is replaced with ‘ 3 -power-free’. Furthermore, the number 7 3 is best possible.

متن کامل

Acoustic Wave propagation through multilayer Thue-Morse structures containing two, three and four materials of Cu, Al, MgO and Pb

In this paper, the acoustic wave propagation in multilayer Thue-Morse structures has theoretically been studied. The composing layers were assumed do be Cu, Al, Mgo and Pb materials. Two, three or four different materials have been used in a typical structure. By using a perpendicular incident acoustic wave, the transmission coefficient was calculated and then by its old, the frequency gaps was...

متن کامل

On the structure of compacted subword graphs of Thue-Morse words and their applications

We investigate how syntactic properties of Thue-Morse words are related to special type of automata/graphs. The directed acyclic subword graph (dawg, in short) is a useful deterministic automaton accepting all su xes of the word. Its compacted version (resulted by compressing chains of states) is denoted by cdawg. The cdawgs of Thue-Morse words have regular and very simple structure, in particu...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Electr. J. Comb.

دوره 24  شماره 

صفحات  -

تاریخ انتشار 2017