Swapnil Garg - Antipowers in Uniform Morphic Words and the Fibonacci Word

dmtcs:7134 - Discrete Mathematics & Theoretical Computer Science, December 9, 2021, vol. 23, no. 3 - https://doi.org/10.46298/dmtcs.7134
Antipowers in Uniform Morphic Words and the Fibonacci Word

Authors: Swapnil Garg

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.


Volume: vol. 23, no. 3
Section: Combinatorics
Published on: December 9, 2021
Accepted on: October 20, 2021
Submitted on: January 27, 2021
Keywords: Mathematics - Combinatorics,05A05


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