Let $p$ bea a prime number and $|\, \cdot\, |_p$ the $p$-adic absolute value of $\mathbf{Q}$ and on the $p$-adic field $\mathbf{Q}_p$ normalized such that $|p|_p=p^{-1}$. Let $\xi$ be a quadratic real number and $\alpha$ a quadratic $p$-adic number. We prove that there exist positive, effectively computable, real numbers $c_1=c_1(\xi)$, $\tau_1=\tau_1(\xi)$, $c_2=c_2(\xi)$, $\tau_2=\tau_2(\xi)$, such that $$|y\xi - x|\cdot |y|_p \geq c_1 |y|^{-2+\tau_1}, \; \text{for}\; x,y \; \text{in}\; \mathbb{Z}_{\neq 0},$$ and $$|b\alpha -a|_p\geq c_2 |ab|^{-2+\tau_2}, \; \text{for}\; a,b\; \text{in}\; \mathbb{Z}_{\neq 0}.$$ Both results improve the effective lower bounds which follow from an easy Liouville-type argument.
| On Effective Approximation to Quadratic Numbers | 552.8KB |