2017/05/17 by Narayanan, Shyam
#05A05 #68R15 #Combinatorics (math.CO) #FOS: Mathematics #G.2.1
paper · doi:10.48550/arxiv.1705.06310
We say that a word w of length kn is a k-antipower if it can be written in the form w1 ⋯ wk, where each wi is a distinct word of length n. We analyze prefixes of the Thue-Morse word t and lengths of antipowers occurring in them. Define Γ(k) to be the largest odd n such that the prefix of t of length kn is not a k-antipower, and γ(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 γ(k) and Γ(k)-γ(k). Our bounds on γ(k) affirmatively answer one conjecture of Defant and make substantial progress towards answering a second conjecture of Defant. It was previously known that Γ(k) and γ(k) grow linearly in k, but our bounds on Γ(k)-γ(k) prove that Γ(k)-γ(k) also grows linearly in k.