Thue–Morse Constant is Not Badly Approximable
Dzmitry Badziahin, Evgeniy Zorin · International Mathematics Research Notices · 2014
We prove that Thue–Morse constant |${\tau _{\rm TM}}=0.01101001\dots _2$| is not a badly approximable number. Moreover, we prove that |${\tau _{\rm TM}}(a)=0.01101001\dots _a$| is not badly approximable for every integer base |$a\geq 2$| such that |$a$| is not divisible by 15. At the same time, we provide a precise formula for convergents of the Laurent series |${\tilde {f}_{\rm TM}}(z) = z^{-1}\prod _{n=1}^\infty (1-z^{-2^n})$|, thus developing further the research initiated by Alf van der Poorten and others.