On the sequence of fractional parts of the ratio of Fibonacci numbers $x_{n+1}=\left\{\frac{F_{n+1}}{F_n}x_n\right\}$

In this paper for the expension of real numbers on Fibonacci sequence theorems on the uniform distribution of remainders for almost of all real numbers in the sense of Lebesgue's measure. the conclusion of this theorem is based on the Weyl's criteria of the uniform distribution of a sequence modulo unit and on the lemma.