Dual Group Structures in Diophantine Approximations

As revealed by the MLA(Mesopotamian Logarithm Algorithm) for logarithmic convergents, a similar property appears in other irrationals when analyzed in their corresponding space.

Logarithm Case Recap:

Irrational: \(\alpha = \log_b(a)\)

Convergent: \(p/q \approx \log_b(a) \Rightarrow q \times log_b(a) \approx p \Rightarrow a^q \approx b^p\)

Sequence: \(r_x = a^x \times b^{y_x}\) reduced to \([1, b)\). This is like looking at \(a^x\) "modulo \(b\)" multiplicatively. \(y_x\) tracks the 'overflow' exponent of \(b\). (This highlights the absence of a standard shorthand notation for multiplicative modulus; see link)

Sorted Sequence: Sorting \(r_x\) for \(x=1\ldots q\) gives indices \(x_k\).

Structure: \(x_k\) forms \(\mathbb{Z}/q\mathbb{Z}\) (gen \(p^{-1} \mod q\)), \(y_{x_k}\) forms \(q\) terms of \(\mathbb{Z}/p\mathbb{Z}\) (gen \(q^{-1} \mod p\)).


Trigonometric Case (Angle)

Irrational: We need an irrational quantity related to the angle. Let's use \(\alpha = \theta / (2\pi)\). (assuming \(\theta\) is not a rational multiple of \(2\pi\)).

Convergent: \(p/q \approx \theta / (2\pi) \Rightarrow q \times \theta / (2\pi) \approx p \Rightarrow q\theta ≈ 2\pi p\). This means \(q\) rotations by \(\theta\) is close to \(p\) full \(2\pi\) rotations.

Sequence: What's the equivalent of \(a^x \mod 1:b\)? The natural analogue for angles is \(x\theta \mod 2\pi\). Let \(r_x = (x\theta) \pmod{2\pi}\). This sequence lives in \([0, 2\pi)\).

What is \(y_x\) ? It's the number of full rotations removed: \(xθ = y_x \times 2\pi + r_x\). So, \(y_x = \lfloor x\theta / (2\pi)\rfloor\).

Sorted Sequence: Sort \(r_x\) for \(x=1\ldots q\) to get indices \(x_k\).

Structure: \(x_k\) forms \(\mathbb{Z}/q\mathbb{Z}\) (gen \(p^{-1} \mod q\)), \(y_{x_k}\) forms \(q\) terms of \(\mathbb{Z}/p\mathbb{Z}\) (gen \(q^{-1} \mod p\)).

This directly mimics the log case by replacing the multiplicative group \((\mathbb{R}^+, \cdot)\) modulo \(b\) with the additive group \(\mathbb{R} \mod 2\pi\) (the circle group \(S^1\)). The relationship \(q\theta \approx 2\pi p\)  is the direct analogue of \(a^q \approx b^p\). The Three Gap Theorem describes the structure of the sorted \(r_x\) values (the points \(x\theta \mod 2\pi\) on the circle), and their ordering is intimately linked to the continued fraction convergents \(p/q\). The generators likely arise from the relationship \(q(p'/q') - p(q'/q') = \pm \)1 between consecutive convergents.

(Need to test which inverse/element works. The structure \(p_{n-1} q_n - p_n q_{n-1} = (-1)^n\) from continued fractions is key here, likely determining the specific generators.)