Proof of a conjecture of Krawchuk and Rampersad
Jeffrey Shallit
TL;DR
This paper proves a 2018 conjecture regarding the extremal behavior of the function counting length-$n$ factors of the Thue-Morse word up to cyclic rotation, advancing understanding of its combinatorial properties.
Contribution
The paper provides a rigorous proof of a conjecture about the extremal behavior of factor counts in the Thue-Morse sequence, a significant open problem in combinatorics on words.
Findings
Confirmed the conjecture on extremal behavior of $c(n)$
Characterized the maximum and minimum values of $c(n)$
Enhanced understanding of the structure of the Thue-Morse word
Abstract
We prove a 2018 conjecture of Krawchuk and Rampersad on the extremal behavior of , where counts the number of length- factors of the Thue-Morse word , up to cyclic rotation.
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Taxonomy
TopicsAnalytic Number Theory Research · semigroups and automata theory · Quasicrystal Structures and Properties