Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Functions on Antipower Prefix Lengths of the Thue-Morse Word   We say that a word $w$ of length $kn$ is a $k$-\textit{antipower} if it can
be written in the form $w_1 \cdots w_k$, where each $w_i$ is a distinct word of
length $n$. We analyze prefixes of the Thue-Morse word $\textbf{t}$ and lengths
of antipowers occurring in them. Define $\Gamma(k)$ to be the largest odd $n$
such that the prefix of $\textbf{t}$ of length $kn$ is not a $k$-antipower, and
$\gamma(k)$ to be the smallest odd $n$ such that the corresponding prefix is a
$k$-antipower. We provide strong bounds on the asymptotic values of $\gamma(k)$
and $\Gamma(k)-\gamma(k)$. Our bounds on $\gamma(k)$ affirmatively answer one
conjecture of Defant and make substantial progress towards answering a second
conjecture of Defant. It was previously known that $\Gamma(k)$ and $\gamma(k)$
grow linearly in $k$, but our bounds on $\Gamma(k)-\gamma(k)$ prove that
$\Gamma(k)-\gamma(k)$ also grows linearly in $k$.

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The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex