The best two-term underapproximation using numbers from Fibonacci-type sequences

Abstract

This paper studies the greedy two-term underapproximation of $\theta\in (0,1]$ using reciprocals of numbers from a Fibonacci-type sequence $(c_n)_{n=1}^\infty$. We find the set of $\theta$ whose greedy two-term underapproximation is the best among all two-term underapproximations using $1/c_n$'s. We then derive a neat description of the set when $(c_n)_{n=1}^\infty$ is the Fibonacci sequence or the Lucas sequence.