We give local upper and lower bounds for the eigenvalues of the modular operator associated to an ergodic action of a compact quantum group on a unital C*-algebra. They involve the modular theory of the quantum group and the growth rate of quantum dimensions of its representations and they become sharp if other integral invariants grow subexponentially. For compact groups, this reduces to the finiteness theorem of Hoegh-Krohn, Landstad and Stormer. Consequently, compact quantum groups of Kac type admitting an ergodic action with a non-tracial invariant state must have representations whose dimensions grow exponentially. In particular, S_{-1}U(d) acts ergodically only on tracial C*-algebras. For quantum groups with non-involutive coinverse, we derive a lower bound for the parameters 0<lambda<1 of factors of type III_lambda that can possibly arise from the GNS representation of the invariant state of an ergodic action with a factorial centralizer.
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