On the rational approximations to the real numbers corresponding to the differences of the Fibonacci sequence

TL;DR

This paper provides a new proof for the irrationality exponents of Fibonacci-related real numbers and determines all such exponents for differences within the Fibonacci sequence.

Contribution

It introduces a novel proof technique based on Fibonacci sequence structure and fully characterizes the irrationality exponents for Fibonacci differences.

Findings

New proof for irrationality exponents of Fibonacci numbers

Complete characterization of irrationality exponents for Fibonacci differences

Enhanced understanding of Fibonacci sequence's approximation properties

Abstract

Based on the structure of Fibonacci sequence, we give a new proof for the irrationality exponents of the Fibonacci real numbers. Moreover, we obtain all the irrationality exponents of the real numbers corresponding to the differences of Fibonacci sequence.