关于间隙原理的推广
数论
2022-06-29 v2
摘要
设 α \alpha α d ≥ 3 d \geq 3 d ≥ 3 β ∈ Q ( α ) \beta \in \mathbb Q(\alpha) β ∈ Q ( α ) μ \mu μ ( d / 2 ) + 1 < μ < d (d/2) + 1 < \mu < d ( d /2 ) + 1 < μ < d C 0 C_0 C 0 α \alpha α β \beta β μ \mu μ C 0 C_0 C 0 C 1 C_1 C 1 C 2 C_2 C 2 x 1 / y 1 x_1/y_1 x 1 / y 1 x 2 / y 2 x_2/y_2 x 2 / y 2 H ( x 2 , y 2 ) ≥ H ( x 1 , y 1 ) ≥ C 1 H(x_2, y_2) \geq H(x_1, y_1) \geq C_{1} H ( x 2 , y 2 ) ≥ H ( x 1 , y 1 ) ≥ C 1 ∣ α − x 1 y 1 ∣ < C 0 H ( x 1 , y 1 ) μ , ∣ β − x 2 y 2 ∣ < C 0 H ( x 2 , y 2 ) μ , \left|\alpha - \frac{x_1}{y_1}\right| < \frac{C_0}{H(x_1, y_1)^\mu}, \quad \left|\beta - \frac{x_2}{y_2}\right| < \frac{C_0}{H(x_2, y_2)^\mu}, α − y 1 x 1 < H ( x 1 , y 1 ) μ C 0 , β − y 2 x 2 < H ( x 2 , y 2 ) μ C 0 , H ( x 2 , y 2 ) > C 2 − 1 H ( x 1 , y 1 ) μ − d / 2 H(x_2, y_2) > C_{2}^{-1} H(x_1, y_1)^{\mu - d/2} H ( x 2 , y 2 ) > C 2 − 1 H ( x 1 , y 1 ) μ − d /2 s , t , u , v s, t, u, v s , t , u , v s v − t u ≠ 0 sv - tu \neq 0 s v − t u = 0 β = s α + t u α + v 且 x 2 y 2 = s x 1 + t y 1 u x 1 + v y 1 , \beta = \frac{s\alpha + t}{u\alpha + v} \quad \text{且} \quad \frac{x_2}{y_2} = \frac{sx_1 + ty_1}{ux_1 + vy_1}, β = uα + v s α + t 且 y 2 x 2 = u x 1 + v y 1 s x 1 + t y 1 , H ( x , y ) = max ( ∣ x ∣ , ∣ y ∣ ) H(x, y) = \max(|x|, |y|) H ( x , y ) = max ( ∣ x ∣ , ∣ y ∣ ) x / y x/y x / y μ − d / 2 \mu - d/2 μ − d /2 α \alpha α β \beta β x 1 / y 1 x_1/y_1 x 1 / y 1 x 2 / y 2 x_2/y_2 x 2 / y 2 α \alpha α β \beta β p p p
引用
@article{arxiv.2112.13919,
title = {On the Generalization of the Gap Principle},
author = {Anton Mosunov},
journal= {arXiv preprint arXiv:2112.13919},
year = {2022}
}