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ON GENERALIZED THUE–MORSE FUNCTIONS AND THEIR VALUES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 108 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 11 March 2019, pp. 177-201
- Print publication:
- April 2020
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- Article
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In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products
$f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$, 
$d\in \mathbb{N}$, 
$d\geq 2$, which generalize the generating function 
$f_{2}(x)$ of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of 
$x^{-d+1}f_{d}(x)$ have a regular structure. We also address the question of whether the corresponding Mahler numbers 
$f_{d}(a)\in \mathbb{R}$, 
$a,d\in \mathbb{N}$, 
$a,d\geq 2$, are badly approximable.
