On the Generalization of the Gap Principle

Abstract

Let $\alpha$ be a real algebraic number of degree $d \geq 3$ 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 $$ H(x_2, y_2) \geq H(x_1, y_1) \geq C_{1} $$ and $$ \left|\alpha - \frac{x_1}{y_1}\right| < \frac{C_0}{H(x_1, y_1)^\mu}, \quad \left|\beta - \frac{x_2}{y_2}\right| < \frac{C_0}{H(x_2, y_2)^\mu}, $$ 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 $sv - tu \neq 0$, such that $$ \beta = \frac{s\alpha + t}{u\alpha + v} \quad \text{and} \quad \frac{x_2}{y_2} = \frac{sx_1 + ty_1}{ux_1 + vy_1}, $$ or both. Here $H(x, y) = \max(|x|, |y|)$ is the height of $x/y$. Since $\mu - d/2$ exceeds one, 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.