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Continued fractions of certain Mahler functions. (English) Zbl 1432.11089
Let $$P(x)=1+u_1x+\cdots+u_{d-1}x^{d-1}\in{\mathbb{F}}[x]$$ 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.

##### MSC:
 11J82 Measures of irrationality and of transcendence 11B85 Automata sequences 11J04 Homogeneous approximation to one number 11J70 Continued fractions and generalizations
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##### References:
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