On a Simultaneous Approximation Problem concerning Binary Recurrences

Let Rn (n = 0, 1, 2, . . .) be a second order linear recursive sequence of rational integers defined by Rn = ARn−1+BRn−2 for n > 1, where A and B are integers and the initial terms are R0 = 0, R1 = 1. It is known, that if α, β are the roots of the equation x2 −Ax−B = 0 and |α| > |β|, then Rn+1/Rn −→ α as n −→ ∞. Approximating α with the rational number Rn+1/Rn, it was shown that ∣∣∣α− Rn+1 Rn ∣∣∣ < 1 c·|Rn|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 α and αs by the rational numbers Rn+1/Rn and Rn+s/Rn simultaneously.