Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 17, No. 2, pp. 71-76 (2001)

On a simultaneous approximation problem concerning binary recurrences

Péter Kiss

Department of Mathematics, Károly Eszterházy College, H-3301 Eger, P.O.Box 43, Hungary

Abstract: Let $R_n (n=0, 1, 2, \ldots )$ be a second order linear recursive sequence of rational integers defined by $R_n=AR_{n-1}+BR_{n-2}$ for $n>1$, where $A$ and $B$ are integers and the initial terms are $R_0=0, R_1=1$. It is known, that if $\alpha, \beta$ are the roots of the equation $x^2-Ax-B=0$ and $|\alpha|>|\beta|$, then $R_{n+1}/R_n\longrightarrow \alpha$ as $n\longrightarrow \infty$. Approximating $\alpha$ with the rational number $R_{n+1}/R_n$, it was shown that $\left|\alpha -{{R_{n+1}}\over R_n}\right|<{1\over {c\cdot |R_n|^2}}$ holds with a constant $c>0$ for infinitely many $n$ if and only if $|B|=1$. In this paper we investigate the quality of the approximation of $\alpha$ and $\alpha ^s$ by the rational numbers $R_{n+1}/R_n$ and $-R_{n+s}/R_n$ simultaneously.

Keywords: Linear recursive sequence, Diophantine approximation.

Classification (MSC2000): 11B37; 11J13, 11J86.

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