On effective approximation to quadratic numbers

Let p be a prime number and | · |p the p-adic absolute value on Q and on the p-adic field Qp normalized such that |p|p = p −1 . Let ξ be a quadratic real number and α a quadratic p-adic number. We prove that there exist positive, effectively computable, real numbers c1 = c1(ξ), τ1 = τ1(ξ), c2 = c2(α), τ2 = τ2(α), such that |yξ − x| · |y|p ≥ c1|y| −2+τ1 , for x, y ∈ Z̸=0, and |bα − a|p ≥ c2|ab| −2+τ2 , for a, b ∈ Z̸=0. Both results improve the effective lower bounds which follow from an easy Liouville-type argument.