bibkey: garg2021antipowers authors: Swapnil Garg year: 2021 title: “Antipowers in Uniform Morphic Words and the Fibonacci Word” doi: 10.46298/dmtcs.7134 url: https://arxiv.org/abs/1907.10816v4 claim: ‘Let $F_n$ be an even Fibonacci number. Then, there is an $(F_n-1)$-antipower with block length $\frac{F_n}{2}+F_{n-1}$ that is a prefix of $\mathbf{f}$.’ strata_touched:

• D5/S1/Words/Antipowers/GargFibonacciPrefixAntipower license: citation-only triage: anchor

Fibonacci prefix antipowers

Verified locator

DOI: https://doi.org/10.46298/dmtcs.7134

Source: https://arxiv.org/abs/1907.10816v4

DMTCS 23(3), article 14. Section 3 contains Fact 14, Lemma 15, Propositions 16–17 and Conjecture 18. Conjecture 18 is on printed page 8.

Source statement

Conjecture 18 states verbatim:

Let $F_n$ be an even Fibonacci number. Then, there is an $(F_n-1)$-antipower with block length $\frac{F_n}{2}+F_{n-1}$ that is a prefix of $\mathbf{f}$.

The source’s definition in §3 (printed page 5) states verbatim:

We prove that the Fibonacci word $\mathbf{f}$, which is equal to ${\phi }^{\omega }(0)$ for $\phi (0)=01$, $\phi (1)=0$ and thus pure morphic but not generated by a uniform morphism, also satisfies Conjecture 3.

The abstract (printed page 1) states verbatim:

Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length.

The Fibonacci word is the fixed point beginning with 0 of the morphism $0\mapsto 01$, $1\mapsto 0$. The Fibonacci numbers satisfy $F_1=F_2=1$. A $k$-antipower consists of $k$ consecutive pairwise distinct blocks of equal length.

The encoding uses letters in Fin 2, the recursive prefixes $S_0=[0]$, $S_1=[0,1]$, $S_{k+2}=S_{k+1}{S}_k$, and their compatible infinite limit. Boolean true in the golden-word model corresponds to letter 0. Nat.fib has the source’s indexing, and Nat.div by 2 is exact whenever $F_n$ is even. Blocks begin at $jL$, with $0\le j<F_n-1$ and $L=F_n/2+F_{n-1}$.