? hence, in particular, that ? |log(q)| ? CΘ ・ |log(q)| for any CΘ ∈ R such that ? |log(Θ)| ? CΘ・|log(q)|. Since [one verifies immediately that] |log(q)| ∈ R is positive, we thus conclude that CΘ ? ?1, as desired. In this context, it is useful to recall that the above argument depends, in an essential way [cf. the discussion of (ii), (vi)], on the theory of [EtTh], which does not admit any evident generalization to the case of N-th tensor powers of Θ-pilot objects, for N ? 2. That is to say, the log-volume of such an N-th tensor power of a Θ-pilot object must always be computed as the result of multiplying the log-volume of the original Θ-pilot object by N ? cf. Remark 2.1.1, (iv); [IUTchII], Remark 3.6.4, (iii), (iv). In particular, although the analogue of the above argument for such an N-th tensor power would lead to sharper inequalities than the inequalities obtained here, it is difficult to see how to obtain such sharper inequalities via a routine generalization of the above argument. In fact, as we shall see in [IUTchIV], these sharper inequalities are known to be false [cf. [IUTchIV], Remark 2.3.2, (ii)].
