Multiples of Golden-Ratio “Modulo” 1: Alternate Form

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For any integer n > 0, the expression F_nϕ−F_n+1 can be repeatedly expanded as below:

Fn ϕ − Fn+1 = Fnϕ − Fn− 1 − Fn = Fn(ϕ − 1)− Fn −1 F = --n− Fn −1 (since ϕ − 1 = 1∕ϕ) (ϕ ) = − 1- (F ϕ − F ) (now, repeatedly expand as above) ϕ n−1 n ( 1 )2 = − -- (Fn −2ϕ − Fn−1) ( ϕ ) 1- 3 = − ϕ (Fn −3ϕ − Fn−2) ... ... ( 1 )n = − -- (F0ϕ − F1) (ϕ )n = − − 1- (4) ϕ

So, the three gap-lengths in corollary 1 can also be expressed as:

--1- |Ft+2ϕ − Ft+3| = ϕt+2 1 |Ft+1ϕ − Ft+2| = ϕt+1 1 |Ftϕ − Ft+1| =--t (5) ϕ