The best two-term underapproximation using numbers from Fibonacci-type sequences
Mark Shiliaev
TL;DR
This work analyzes the problem of two-term underapproximation of $θ$ by sums of reciprocals from a Fibonacci-type sequence $(a_n)$ with $a_n=a_{n-1}+a_{n-2}$ and $χ>0$. It derives a precise necessary-and-sufficient condition for the greedy two-term underapproximation $\mathcal{G}(θ)$ to be optimal among all two-term underapproximations, described via a union of intervals and a parameter $\xi(n)$ tied to the recurrence. The condition is then specialized to the Fibonacci and Lucas sequences, yielding explicit interval descriptions and proving that the corresponding $\xi(n)$ values satisfy $\xi(n)=4n+4$ (Fibonacci) and $\xi(n)=4n+6$ (Lucas). The results connect greedy Egyptian-fraction-type approximations with exact interval characterizations and open paths to extensions to broader recurrences and measure-theoretic questions.
Abstract
This paper studies the greedy two-term underapproximation of $θ\in (0,1]$ using reciprocals of numbers from a Fibonacci-type sequence $(c_n)_{n=1}^\infty$. We find the set of $θ$ 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.