Abstract
We prove the rank of the group of signatures of the circular units (hence also the full group of units) of ${\mathbb Q}( \zeta_m)^+$ tends to infinity with $m$. We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime $p$ the signature rank of the units of ${\mathbb Q}( \zeta_{p^n})^+$ differs from $\varphi(p^n)/2$ by an amount that is bounded independent of $n$. Finally, we show conditionally that for general cyclotomic fields the unit signature rank can differ from its maximum possible value by an arbitrarily large amount.
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