On the possible quantities of Fibonacci numbers that occur in some type of intervals

Abstract

In this paper, we show that for any integer $a \geq 2$, each of the intervals $[a^k , a^{k + 1})$ ($k \in \mathbb{N}$) contains either $\left\lfloor \frac{\log a}{\log\Phi}\right\rfloor$ or $\left\lceil \frac{\log a}{\log\Phi}\right\rceil$ Fibonacci numbers. In addition, the density (in $\mathbb{N}$) of the set of the all natural numbers $k$ for which the interval $[a^k , a^{k + 1})$ contains exactly $\left\lfloor \frac{\log a}{\log\Phi}\right\rfloor$ Fibonacci numbers is equal to $\left(1 - \left\langle \frac{\log a}{\log\Phi}\right\rangle\right)$ and the density of the set of the all natural numbers $k$ for which the interval $[a^k , a^{k + 1})$ contains exactly $\left\lceil \frac{\log a}{\log\Phi}\right\rceil$ Fibonacci numbers is equal to $\left\langle \frac{\log a}{\log\Phi}\right\rangle$.