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.)