Some remarks on the $\ell$-adic regulator. III

Let $K$ be a finite extension of the field of rational $\ell$-adic numbers $\mathbb Q_\ell$, and let $K_\infty$ be the cyclotomic $\mathbb Z_\ell$-extension of $K$. For an intermediate field $K_n$ in $K_\infty/K$, let $U(K_n)$ be the group of units of $K_n$ and put $U(K_n)^\perp=\{x\in K_n\mid\operatorname{Sp}_{K_n/\mathbb Q_\ell}(x\log u)\in {\mathbb Z}_\ell$ for all $u\in U(K_n)\}$, where $\log\colon U(K_n)\to K_n$ is the $\ell$-adic logarithm. We give an approximate characterization of $U(K_n)^\perp$. The proofs are based on the use of Laurent series with integer coefficients and infinite principal part.