On the maximal number of cubic subwords in a string

TL;DR

This paper explores the maximum number of cubic subwords in a string, providing new bounds and exact counts for squares and cubes, advancing understanding of repetitive patterns in words.

Contribution

It introduces new bounds for the maximum number of cubic subwords, improving previous estimates and offering exact counts for certain types of squares and cubes.

Findings

Maximum number of nonprimitive squares is exactly ⌊n/2⌋ - 1.

Maximum number of subwords of the form x^k, k≥3, is exactly n-2.

Maximum number of cubes in a word of length n is between n/2 and 4n/5.

Abstract

We investigate the problem of the maximum number of cubic subwords (of the form $www$) in a given word. We also consider square subwords (of the form $ww$). The problem of the maximum number of squares in a word is not well understood. Several new results related to this problem are produced in the paper. We consider two simple problems related to the maximum number of subwords which are squares or which are highly repetitive; then we provide a nontrivial estimation for the number of cubes. We show that the maximum number of squares $xx$ such that $x$ is not a primitive word (nonprimitive squares) in a word of length $n$ is exactly $\lfloor \frac{n}{2}\rfloor - 1$, and the maximum number of subwords of the form $x^k$, for $k\ge 3$, is exactly $n-2$. In particular, the maximum number of cubes in a word is not greater than $n-2$ either. Using very technical properties of occurrences of cubes, we improve this bound significantly. We show that the maximum number of cubes in a word of length $n$ is between $(1/2)n$ and $(4/5)n$. (In particular, we improve the lower bound from the conference version of the paper.)