Anti-Powers in Primitive Uniform Substitutions

TL;DR

This paper extends the known existence of bounded anti-powers in fixed points of uniform primitive morphisms from binary alphabets to arbitrary finite alphabets, using recognisability techniques.

Contribution

It generalizes previous results on anti-powers in primitive uniform morphic words to broader alphabets, employing recognisability methods.

Findings

Existence of a constant C for anti-powers in fixed points of uniform primitive morphisms on any finite alphabet.

Extension of previous binary results to arbitrary finite alphabets.

Independent proof by S. Garg using different techniques.

Abstract

In a recent work, A. Berger and C. Defant showed that if $x$ is a fixed point of a binary uniform and primitive morphism, then there exists a constant $C$ such that for all positive integers $i,k,$ beginning in position $n$ in $x$ is a $k$-anti-power with block length at most $Ck$. They ask whether this result extends to a broader class of morphic words. In this note we extend their results to fixed points of uniform primitive morphisms on arbitrary finite alphabets. Our methods make use of the recognisability of uniform primitive morphisms. This result was proved independantly by S. Garg, using a different technique.