Multiples of Golden-Ratio “Modulo” 1: The Three Gap-Lengths

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The Three Gap-Lengths

We will be using a few facts from the theory of continued fractions. Say, p_k∕q_k represent the convergents from the continued fraction of a real number α > 0. Then, it is known that for all k ≥ 0:

|| || |α − pk| < ---1-- | qk| qkqk+1 --1- ⇔ |qkα− pk| < qk+1

For all k ≥ 0, since q_k+1 ≥ 1, so:

|qkα− pk| < 1

It is also known that, for all k ≥ 0:

| | α − pk-= ||α − pk||(− 1)k qk | qk| ⇔ q α − p = |q α− p |(− 1 )k k k k k

Due to above relations, and the fact that any integer can be added or subtracted inside {}, {q_kα} can also be written as:

{qkα} = {qkα − pk} = {1 + qkα − pk} = {1 + |qkα − pk|(− 1)k} = qkα − pk (for even k) {1 − (pk − qkα )} = 1 − (pk − qkα) (for odd k) (3)

Now we compute the gap-lengths for ϕ. As deﬁned in section “Three Distance Theorem” of [1], we will use u₁ and u_N to refer to the two “corner points”.

Consider the case when t is even. Using theorem 4 in [1], the u₁ and u_N for ϕ will be:

u1 = qt = Ft+1 uN = qt− 1 + rqt = Ft + (1)Ft+1 = Ft+2

We noted earlier, due to relation (2), that F_t+2 is the largest Fibonacci number not exceeding N. Now we see that the two corner points correspond to the two largest Fibonacci numbers not exceeding N.

Now we use theorem 2 in [1] to ﬁnd the three gap-lengths, which are speciﬁed as L₁, L₂ and L₁ + L₂ in that theorem.

L1 = {u1α } = {qtα} = qtα − pt (due to (3) with t even) = Ft+1ϕ − Ft+2 L = 1 − {u α } 2 N = 1 − {qt+1α } (since Ft+2 is also qt+1) = 1 − (1− (pt+1 − qt+1α)) (due to (3) with t + 1 odd ) = pt+1 − qt+1α = Ft+3 − Ft+2ϕ

L1 + L2 = Ft+1ϕ − Ft+2 + Ft+3 − Ft+2ϕ = Ft+1 − Ftϕ

This was for even t. Similarly, it can be proved that for t as odd, the expressions of the three lengths will be:

L1 = − (Ft+3 − Ft+2ϕ) L2 = − (Ft+1ϕ − Ft+2) L1 + L2 = − (Ft+1 − Ftϕ)

Corollary 1. The two smallest gap-lengths (L₁ and L₂, in this order or reverse) for α = ϕ are:

|Ft+2ϕ− Ft+3 | and |Ft+1ϕ− Ft+2 |

where, F_t+2 (t ≥ 0) is the largest Fibonacci number not exceeding N. The gap-length L₁ + L₂ can also be written as:

|Ftϕ− Ft+1 |.

Note how u₁ and u_N, and hence the three gap-lengths, change only when t changes. Due to the relation (2) which deﬁnes t, that occurs whenever N attains any Fibonacci number.

Note that, gaps of length L₁ + L₂ need not exist for all N. From corollary 5 in [1], gap-length L₁ + L₂ will not exist if s = q_t − 1. For case of ϕ, that can be rewritten as:

s = qt − 1 ⇔ N − qt−1 − rqt = qt − 1 (due to relation (4) in [1 ]) ⇔ N − Ft − Ft+1 = Ft+1 − 1 ⇔ N = Ft+3 − 1

Corollary 2. For α = ϕ, the gap-length of L₁+L₂ does not exist whenever N + 1 is some Fibonacci number.

So, in Figure 1 for N = 12, which is F₇ − 1, we should expect only two distinct gap-lengths.

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