Are irrational numbers well-approximated over $ p$ -adic numbers? For example let $ x = \sqrt{2}$ then $ x \notin \mathbb{Q}_3$ . Can we have infinitely many rationals $ \frac{m}{n}\in \mathbb{Q}$ such that: $ $ \left|\sqrt{2} – \frac{m}{n}\right|_3 < |n|_3 $ $ What happens if we put more than one place. Could we have two simultaneous inequalities? $ $ \left|\sqrt{2} – \frac{m}{n}\right|_3 < |n|_3 \text{ and } \left|\sqrt{2} – \frac{m}{n}\right|_5 < |n|_5 $ $ By weak approximation, we can find rational numbers which are arbitrarily close over both places: $ $ \left|\sqrt{2} – \frac{m}{n}\right|_3 < \epsilon_1 \text{ and } \left|\sqrt{2} – \frac{m}{n}\right|_5 < \epsilon_2 $ $ but I am asking of $ x = \sqrt{2}$ can be well-approximated in both places, or what’s the best exponent we can obtain?