On the generalization of the gap principle

Abstract

Let \(\alpha \) be a real algebraic number of degree \(d \ge 3\) and let \(\beta \in {\mathbb {Q}}(\alpha )\) be irrational. Let \(\mu \) be a real number such that \((d/2) + 1< \mu < d\) and let \(C_0\) be a positive real number. We prove that there exist positive real numbers \(C_1\) and \(C_2\), which depend only on \(\alpha \), \(\beta \), \(\mu \) and \(C_0\), with the following property. If \(x_1/y_1\) and \(x_2/y_2\) are rational numbers in lowest terms such that $$\begin{aligned} H(x_2, y_2) \ge H(x_1, y_1) \ge C_{1} \end{aligned}$$and $$\begin{aligned} \left| \alpha - \frac{x_1}{y_1}\right|< \frac{C_0}{H(x_1, y_1)^\mu }, \quad \left| \beta - \frac{x_2}{y_2}\right| < \frac{C_0}{H(x_2, y_2)^\mu }, \end{aligned}$$then either \(H(x_2, y_2) > C_{2}^{-1} H(x_1, y_1)^{\mu - d/2}\), or there exist integers s, t, u, v, with \(sv - tu \ne ~0\), such that $$\begin{aligned} \beta = \frac{s\alpha + t}{u\alpha + v} \quad \text {and} \quad \frac{x_2}{y_2} = \frac{sx_1 + ty_1}{ux_1 + vy_1}, \end{aligned}$$or both. Here \(H(x, y) = \max (|x|, |y|)\) is the height of x/y. Since \(\mu - d/2\) exceeds 1, our result demonstrates that, unless \(\alpha \) and \(\beta \) are connected by means of a linear fractional transformation with integer coefficients, the heights of \(x_1/y_1\) and \(x_2/y_2\) have to be exponentially far apart from each other. An analogous result is established in the case when \(\alpha \) and \(\beta \) are p-adic algebraic numbers.