Thue–Morse Constant is Not Badly Approximable

We prove that Thue–Morse constant is not a badly approximable number. Moreover, we prove that τTM(a)=0.01101001…aτTM(a)=0.01101001…a is not badly approximable for every integer base a≥2 such that a is not divisible by 15. At the same time, we provide a precise formula for convergents of the Laurent series f(z)=z−1∏∞n=1(1−z−2n)f~TM(z)=z−1∏n=1∞(1−z−2n), thus developing further the research initiated by Alf van der Poorten and others.