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ON GENERALIZED THUE–MORSE FUNCTIONS AND THEIR VALUES

  • DZMITRY BADZIAHIN (a1) and EVGENIY ZORIN (a2)

Abstract

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.

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Dzmitry Badziahin acknowledges the support of EPSRC Grant EP/E061613/1. Evgeniy Zorin acknowledges the support of EPSRC Grant EP/M021858/1.

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