## On the length of the continued fraction for values of quotients of power sums.(English)Zbl 1159.11021

More than 30 years ago, Mendès-France asked whether it is true that for coprime integers $$a>1,~b>1$$, the length of the continued fraction of $$(a/b)^n$$ tends to infinity with $$n$$. Motivated by this question, the authors consider the following more general problem. Let $${\mathcal E}$$ be the ring of all power sums, which are functions defined on $${\mathbb N}$$ with values in $${\mathbb Q}$$ given by $\alpha(n)=c_1a_1^n+\cdots+c_ra_r^n,$ where $$a_1>a_2>\cdots>a_r$$ are natural numbers and $$c_1,\ldots,c_r$$ are nonzero rationals. Let $$a_1=\ell(\alpha)$$ be the dominant root of $$\alpha(n)$$. The authors show that under a certain condition on $$\alpha(n)$$ and $$\beta(n)$$ there exist positive integers $$k$$ and $$Q$$ such that given any $$\varepsilon>0$$, the inequality $\Bigl|{{\alpha(n)}\over {\beta(n)}}-{{p}\over {q}}\Bigr|\geq {{1}\over {q^k}}\exp(-\varepsilon n)$ holds for all triples of integers $$(n,p,q)$$ with $$n>0$$ and $$0<q<Q^n$$ with finitely many exceptions. The technical condition is that the assumption $$\ell(\alpha-\zeta \beta)\geq \ell(\beta)$$ must hold for all $$\zeta\in {\mathcal E}$$. This condition is fulfilled, for example, in case of Mendès-France’s question for which one can take $$\alpha(n)=a^n$$ and $$\beta(n)=b^n$$. The authors deduce a number of interesting consequences out of their main result. For example, still under the above condition, the length of the continued fraction of $$\alpha(n)/\beta(n)$$ tends to infinity as $$n\to\infty$$. The proofs rely heavily on a version due to Schlickewei of Schmidt’s Subspace Theorem. The paper concludes with a conjecture asserting that for elements $$\alpha\in {\mathcal E}$$ satisfying suitable conditions the length of the period of the continued fraction of $${\sqrt{\alpha(n)}}$$ should tend to infinity with $$n$$. Meanwhile, this conjecture has been confirmed for power sums satisfying certain conditions in joint work of Y. Bugeaud and the reviewer [Indag. Math. (N.S.) 16, 21–35 (2005; Zbl 1135.11005)], and the general case was dealt with by A. Scremin [Acta Arith. 123, 297–312 (2006; Zbl 1170.11004)].

### MSC:

 11J70 Continued fractions and generalizations 11A55 Continued fractions

### Citations:

Zbl 1135.11005; Zbl 1170.11004
Full Text:

### References:

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