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Equidistribution and generalized Mahler measures. (English) Zbl 1283.37075
Goldfeld, Dorian (ed.) et al., Number theory, analysis and geometry. In memory of Serge Lang. Berlin: Springer (ISBN 978-1-4614-1259-5/hbk; 978-1-4614-1260-1/ebook). 609-638 (2012).
Let \(K\) be a number field or a function field of characteristic \(0\), let \(v\) be a place, and let \(\varphi\) be a rational function defined over \(K\). The generalized Mahler measure of \(F\in K[x]\) with respect to \(\varphi\) and \(v\) is defined to be \[ \int_{\mathbb P^1(\mathbb C_v)} \log |F|_v d\mu_{\varphi, v}, \] where \(\mu_{\varphi, v}\) is a totally \(\varphi\)-invariant probability measure and we use Berkovich \(\mathbb P^1\) for nonarchimedean \(v\). When \(f\) is continuous, \(\int f d\mu_{\varphi, v}\) agrees with the limit as \(n\to \infty\) of the average values of \(f\) at \(\varphi\)-periodic points of period \(n\). On the other hand, \(\log |F|_v\) is in general not continuous, so we cannot directly use this compute the generalized Mahler measure.
The main result (Theorem 5.7) of this article shows that in fact we can calculate the generalized Mahler measure this way, generalizing the fact that the usual Mahler measure can be computed as the limit as \(n\to \infty\) of the average values of \(\log |F|\) at the \(n\)-th roots of unity. As a corollary, the \(\varphi\)-canonical height of an element in \(\overline K\) can be computed as a sum over \(v\) of these averages (Theorem 5.10).
In contrast to the case of evaluating at the roots of unity, the authors use Roth’s theorem instead of linear forms in logarithms. As the authors comment, the crucial Diophantine argument had already appeared in [J. H. Silverman, Duke Math. J. 71, No. 3, 793–829 (1993; Zbl 0811.11052)] without their prior knowledge. When \(F\) is divisible by \(x-\beta\) for a \(\varphi\)-periodic \(\beta\), a more careful argument is necessary to bound the order of vanishing of \(\varphi^k (x) = x\) at \(x = \beta\) as \(k\to \infty\). The article ends with a counterexample to the main theorem when \(F\) has transcendental coefficients, as well as posing what the appropriate “small points” version might be.
For the entire collection see [Zbl 1230.00036].

37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
11J68 Approximation to algebraic numbers
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