Antipowers in Uniform Morphic Words and the Fibonacci Word

Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a wordcomposed of $k$ pairwise distinct, concatenated words of equal length. Bergerand Defant conjecture that for any sufficiently well-behaved aperiodic morphicword $w$, 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 extendtheir result to alphabets of arbitrary finite size and characterize those wordsfor which the result does not hold. We also prove their conjecture in thespecific case of the Fibonacci word.Comment: 8 page