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Consider the multiples of the Golden-Ratio ϕ: ϕ,2ϕ,3ϕ,…. What can we say about the distribution of their fractional-parts in interval [0,1]? In this article we find out how, due to the Three Distance Theorem and properties of the Golden-Ratio, these values are well-distributed.
Let α < 1 be any positive real number and N any positive integer. For any real number x, let {x} denote the fractional part of x, i.e. {x} = x −⌊x⌋. What can we say about distribution of values:
|
| (1) |
in the interval [0,1]?
In an earlier article titled Three Distance Theorem [1], we analyzed such distribution for any general α. The basic statement of the Three Distance Theorem is very simple. It says that, if the values in (1) are sorted and we find the differences of all neighboring values (including interval boundaries 0 and 1), which we can call “gap lengths”, then there are either two or three distinct gap-lengths. We also learned how these gap-lengths are related to the Simple Continued Fraction of α. In this article we will make use of the observations and theorems from [1], for the case of α = ϕ.
We will often refer the values in (1) as {mα} where m is a positive integer. The Simple Continued Fraction will be referred as “Continued Fraction”.
For any positive integers m and n, {m(α + n)} = {mα}. This is the reason that, in [1], it was sufficient to consider only α < 1. For the same reason, our analysis here for ϕ ≈ 1.6180339… also applies exactly to ϕ − 1,ϕ + 1,ϕ + 2 etc.
Figure 1 shows the values of {mϕ} for different values of N. Due to the Three Distance Theorem, we expect to see either two or three distinct gap-lengths.
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