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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
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References:

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