# Antipowers in Uniform Morphic Words and the Fibonacci Word

## TL;DR

This paper investigates antipowers in uniform morphic words, proving a conjecture for the Fibonacci word and extending results to words over larger alphabets, providing a deeper understanding of their combinatorial structure.

## Contribution

The paper extends the conjecture on antipowers to arbitrary finite alphabets and characterizes exceptions, also proving the conjecture specifically for the Fibonacci word.

## Key findings

- Proved the antipower conjecture for the Fibonacci word.
- Extended the conjecture to alphabets of arbitrary finite size.
- Characterized words where the conjecture does not hold.

## Abstract

Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length. Berger and Defant conjecture that for any sufficiently well-behaved aperiodic morphic word $w$, there exists a constant $c$ such that for any $k$ and any index $i$, a $k$-antipower with block length at most $ck$ starts at the $i$th position of $w$. They prove their conjecture in the case of binary words, and we extend their result to alphabets of arbitrary finite size and characterize those words for which the result does not hold. We also prove their conjecture in the specific case of the Fibonacci word.