Extending Dekking's construction of an infinite binary word avoiding abelian $4$-powers

TL;DR

This paper presents a new infinite binary word construction avoiding abelian 4-powers with a critical exponent of 3, introduces an algorithm for checking additive power avoidance in morphic sequences, and provides improved estimates on the number of such words.

Contribution

It extends Dekking's construction to create an infinite binary word avoiding abelian 4-powers and develops an algorithm for additive power avoidance in morphic sequences.

Findings

Constructed an infinite binary word with critical exponent 3 avoiding abelian 4-powers.

Developed an algorithm to determine additive power avoidance in morphic sequences.

Established a lower bound of .172^n for the number of binary words of length n avoiding abelian 4-powers.

Abstract

We construct an infinite binary word with critical exponent 3 that avoids abelian 4-powers. Our method gives an algorithm to determine if certain types of morphic sequences avoid additive powers. We also show that there are $\Omega(1.172^n)$ binary words of length $n$ that avoid abelian 4-powers, which improves on previous estimates.