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

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