We construct a new spectrum of units for a commutative symmetric ringspectrum that detects the difference between a periodic ring spectrum and itsconnective cover. It is augmented over the sphere spectrum. The homotopycofiber of its augmentation map is a non-connected delooping of the usualspectrum of units whose bottom homotopy group detects periodicity. Our approach builds on the graded variant of E-infinity spaces introduced injoint work with Christian Schlichtkrull. We construct a group completion modelstructure for graded E-infinity spaces and use it to exhibit our spectrum ofunits functor as right adjoint on the level of homotopy categories. Theresulting group completion functor is an essential tool for studying ringspectra with graded logarithmic structures.
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