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

TL;DR

This paper investigates the distribution of Fibonacci numbers within exponential intervals, establishing precise counts and densities based on logarithmic relationships involving the golden ratio.

Contribution

It provides a novel analysis of Fibonacci number occurrences in exponential intervals, linking their counts to logarithmic ratios and densities.

Findings

Intervals contain either a fixed number or one more Fibonacci number.

Densities of intervals with specific Fibonacci counts are expressed via fractional parts.

Results connect Fibonacci distribution to logarithmic properties of the base.

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 $\left\langle \frac{\log a}{\log\Phi}\right\rangle$.