Signature Ranks of Units in Cyclotomic Extensions of Abelian Number\n Fields

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