ON A SIMULTANEOUS APPROXIMATION PROBLEM CONCERNING BINARY RECURRENCES

Abstract. Let Rn (n = 0, 1, 2,...) be a second order linear recursive se-quence 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+1Rn ∣∣ ∣ < 1c·|Rn|2 holds with a constant c> 0 for infinitely many n if and only if |B | = 1. In this paper we investi-gate the quality of the approximation of α and αs by the rational numbers Rn+1/Rn and Rn+s/Rn simultaneously