On the Generalization of the Gap Principle

Abstract

Let $α$ be a real algebraic number of degree $d \geq 3$ and let $β\in \mathbb Q(α)$ be irrational. Let $μ$ be a real number such that $(d/2) + 1 < μ< d$ 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 $α$, $β$, $μ$ 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 $$ \left|α- \frac{x_1}{y_1}\right| < \frac{C_0}{H(x_1, y_1)^μ}, \quad \left|β- \frac{x_2}{y_2}\right| < \frac{C_0}{H(x_2, y_2)^μ}, $$ then either $H(x_2, y_2) > C_{2}^{-1} H(x_1, y_1)^{μ- d/2}$, or there exist integers $s, t, u, v$, with $sv - tu \neq 0$, such that $$ β= \frac{sα+ t}{uα+ v} \quad \text{and} \quad \frac{x_2}{y_2} = \frac{sx_1 + ty_1}{ux_1 + vy_1}, $$ or both. Here $H(x, y) = \max(|x|, |y|)$ is the height of $x/y$. Since $μ- d/2$ exceeds one, our result demonstrates that, unless $α$ and $β$ 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 $α$ and $β$ are $p$-adic algebraic numbers.