Avoidability of long 𝑘-abelian repetitions

Abstract

We study the avoidability of long k-abelian-squares and k-abeliancubes on binary and ternary alphabets. For k = 1, these are Mäkelä's questions. We show that one cannot avoid abelian-cubes of abelian period at least 2 in infinite binary words, and therefore answering negatively one question from Mäkelä. Then we show that one can avoid 3-abelian-squares of period at least 3 in infinite binary words and 2-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct k-abelian-squares that must appear in an infinite binary word.

On some generalizations of abelian power avoidability

“…In [24], we answer negatively to (1). Then we modify this question to the following one: Question 4.…”

“…Every infinite binary word contains arbitrarily long abelian squares, while ones exist which avoid squares of period at least 3 [8,25]. We recently showed that one can avoid 3-abelian-squares of period at least 3 over a binary alphabet [24]. It seems natural to ask the following: Question 2.…”

Avoiding Two Consecutive Blocks of Same Size and Same Sum over $\mathbb{Z}^2$

“…Problem 4 ( [14,15]). Can we avoid 2-abelian squares of period at least p on the binary alphabet, for some p ∈ N ?…”

“…Thus the largest 2-abelian squares of h 2 (g 3 (h ω 6 (a))) have a period of at most 11 × 5 + 10 = 65. The value 60 is then obtained by checking all the factors of h 2 (g 3 (h ω 6 (a))) of size at most 65. z The value 60 is probably not optimal (the lower bound from [15] is 2). The easiest way to improve this result would be to improve the upper bound on the period for Mäkelä's question.…”

Regularity of k-Abelian Equivalence Classes of Fixed Cardinality

“…Such a research was initiated in [12], and later continued, e.g., in [4]. Other topics of the k-Abelian equivalence such as k-Abelian repetitions, k-Abelian palindromicity and k-Abelian singletons were studied in [8,10,11,14], respectively.…”