Continued fractions of certain Mahler functions.

*(English)*Zbl 1432.11089Let \(P(x)=1+u_1x+\cdots+u_{d-1}x^{d-1}\in{\mathbb{F}}[x]\) be a polynomial of degree \(<d\) with coefficients in a field \({\mathbb{F}}\), let \(f\in{\mathbb{F}}[[x^{-1}]]\) be the Laurent series
\[
f(x)=\prod_{t=0}^\infty P(x^{-d^t})
\]
and let \(g(x)=x^{-1}f(x)\). Given a convergent of the continued fraction of \(g(x)\), the other produces an infinite chain of other convergents. The author finds relations between the partial quotients which can provide recurrence formulae for them. Assuming \(g(x)\) is badly approximable, he produces explicit formulae for the continued fraction of \(g(x)\) in terms of the coefficients \((u_1,\dots,u_{d-1})\) of \(P\) for \(d=2\) and \(d=3\). Assuming \({\mathbb{F}}={\mathbb{Q}}\), if \(b\ge 2\) is an integer such that \(g(b)\not=0\), he deduces that the irrationality exponent of the real number \(g(b)\) is \(2\) if and only if \(g(x)\) is badly approximable. When \(g(x)\) is not badly approximable, he gives an explicit lower bound \(>2\) for the irrationality exponent of \(g(b)\). The results are complete for \(d=2\), while for \(d=3\) the author is able to cover many but not all values of \((u_1,u_2)\in{\mathbb{Q}}^2\). If \(d=3\) and \((u_1,u_2)\) is either \((u,u^2)\), or \((\pm 2,1)\), or \((s^3,-s^2(s^2+1))\) for some \(s\in {\mathbb{Z}}\), then \(g(x)\) is not badly approximable; the author conjectures that for the other values of \((u_1,u_2)\), \(g(x)\) is badly approximable.

Reviewer: Michel Waldschmidt (Paris)

##### MSC:

11J82 | Measures of irrationality and of transcendence |

11B85 | Automata sequences |

11J04 | Homogeneous approximation to one number |

11J70 | Continued fractions and generalizations |

##### Keywords:

Mahler number; Mahler function; infinite product; continued fraction; Hankel determinant; irrationality exponent##### References:

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