Good elements and metric invariants in B_dR ⁺

Let p be a prime, ℚ_p the field of p-adic numbers and ℚ̄_p a fixed algebraic closure of ℚ_p. B _dR ⁺ is the ring of p-adic periods of algebraic varieties over p-adic fields introduced by Fontaine. For each n one defines a canonical valuation w_n on ℚ̄_p such that B _dR ⁺n/Iⁿ becomes the completion of ℚ̄_p with respect to w_n, where I is the maximal ideal of B_dR ⁺. An element α ∈ ℚ̄ _p* is said to be good at level n if w_n(α) = v(αa) where v denotes the p-adic valuation on ℚ̄_p. The set script G sign_n of good elements at level n is a subgroup of ℚ̄_p*. We prove that each quotient group ℚ̄_p*/script G sign_n is a torsion group and that each quotient script G sign₁/script G sign_n is a p-group. We also show that a certain sequence of metric invariants {l _n(Z)}_n∈ℕ associated to an element Z ∈ B_dR ⁺, is constant.