Anti-powers in infinite words

作者：Fici Gabriele^*; Restivo Antonio; Silva Manuel; Zamboni Luca Q

来源：Journal of Combinatorial Theory - Series A, 2018, 157: 109-119.

DOI：10.1016/j.jcta.2018.02.009

• Anti-power
• Unavoidable regularity
• Infinite word

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. 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 3 is ultimately periodic, while 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.

• 出版日期2018-7