Resumen de Classical and uniform exponents of multiplicative p -adic approximation

Yann Bugeaud, Johannes Schleischitz

• Let p be a prime number and ξ an irrational p-adic number. Its irrationality exponent µ(ξ) is the supremum of the real numbers µ for which the system of inequalities 0 < max{|x|, |y|} ≤ X, |yξ − x|p ≤ X−µ has a solution in integers x, y for arbitrarily large real number X. Its multiplicative irrationality exponent µ×(ξ) (resp., uniform multiplicative irrationality exponent µb×(ξ)) is the supremum of the real numbers µb for which the system of inequalities 0 < |xy| 1/2 ≤ X, |yξ − x|p ≤ X−µb has a solution in integers x, y for arbitrarily large (resp., for every sufficiently large) real number X. It is not difficult to show that µ(ξ) ≤ µ×(ξ) ≤ 2µ(ξ) and µb×(ξ) ≤ 4. We establish that the ratio between the multiplicative irrationality exponent µ× and the irrationality exponent µ can take any given value in [1, 2]. Furthermore, we prove that µb×(ξ) ≤ (5 + √ 5)/2 for every p-adic number ξ.