Small points and free abelian groups

Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F. The absolute logarithmic Weil height, respectively, the Néron–Tate height, induces a norm on F∗ modulo torsion, respectively, on E(F) modulo torsion. The groups F∗ and E(F) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. In this paper, we prove the failure of the converse to this statement by explicitly constructing counterexamples.