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

TL;DR

This paper investigates the optimal two-term underapproximation of numbers in (0,1] using reciprocals of Fibonacci-type sequences, identifying the set of numbers for which the greedy method is optimal.

Contribution

It characterizes the set of numbers where the greedy two-term underapproximation is optimal for Fibonacci-type sequences, including Fibonacci and Lucas sequences.

Findings

Identified the set of θ where greedy underapproximation is optimal.

Derived explicit descriptions for Fibonacci and Lucas sequences.

Provided theoretical insights into two-term underapproximation methods.

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.