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We give local upper and lower bounds for the eigenvalues of the modularoperator associated to an ergodic action of a compact quantum group on a unitalC*-algebra. They involve the modular theory of the quantum group and the growthrate of quantum dimensions of its representations and they become sharp ifother integral invariants grow subexponentially. For compact groups, thisreduces to the finiteness theorem of Hoegh-Krohn, Landstad and Stormer.Consequently, compact quantum groups of Kac type admitting an ergodic actionwith a non-tracial invariant state must have representations whose dimensionsgrow exponentially. In particular, S_{-1}U(d) acts ergodically only on tracialC*-algebras. For quantum groups with non-involutive coinverse, we derive alower bound for the parameters 0<\lambda<1 of factors of type III_\lambda thatcan possibly arise from the GNS representation of the invariant state of anergodic action with a factorial centralizer.

Growth rates of dimensional invariants of compact quantum groups and a theorem of Hoegh-Krohn, Landstad and Stormer