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Sunday, March 30, 2025

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: α=logb(a)

Convergent: p/qlogb(a)q×logb(a)paqbp

Sequence: rx=ax×byx reduced to [1,b). This is like looking at ax "modulo b" multiplicatively. yx tracks the 'overflow' exponent of b. (This highlights the absence of a standard shorthand notation for multiplicative modulus; see link)

Sorted Sequence: Sorting rx for x=1q gives indices xk.

Structure: xk forms Z/qZ (gen p1modq), yxk forms q terms of Z/pZ (gen q1modp).


Trigonometric Case (Angle)

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

Convergent: p/qθ/(2π)q×θ/(2π)pqθ2πp. This means q rotations by θ is close to p full 2π rotations.

Sequence: What's the equivalent of axmod1:b? The natural analogue for angles is xθmod2π. Let rx=(xθ)(mod2π). This sequence lives in [0,2π).

What is yx ? 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.)

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