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On Bilu’s equidistribution theorem. (English) Zbl 1029.11030
Branson, Thomas (ed.), Spectral problems in geometry and arithmetic. NSF-CBMS conference on spectral problems in geometry and arithmetic, Iowa City, IA, USA, August 18-22, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 237, 159-166 (1999).
Recently L. Szpiro, E. Ullmo and S. Zhang [Invent. Math. 127, 337-347 (1997; Zbl 0991.11035)] proved that points of small height on an Abelian variety $$A$$ are equidistributed with respect to the Haar measure on $$A(\mathbb{C})$$. Y. F. Bilu [Duke Math. J. 89, 465-476 (1997; Zbl 0918.11035)] established the analogue of this for algebraic tori. In particular he showed that if $$h(\alpha)$$ is the naive height on $$\overline{\mathbb{Q}}$$, the algebraic closure of $$\mathbb{Q}$$, and if $$\{\alpha_n\}$$ is a sequence of points in $$\overline{\mathbb{Q}}$$ such that $\deg(\alpha_n)\rightarrow \infty, \qquad h(\alpha_n)\rightarrow 0$ as $$n\rightarrow \infty$$, then the discrete measures $\Delta_n= \frac{1}{\deg (\alpha_n)} \sum_{\sigma: \mathbb{Q}(\alpha_n)\rightarrow \mathbb{C}} \delta_{\sigma(\alpha_n)} (x)$ converge weakly (relative to the space of continuous functions on $$\mathbb{C}$$ with compact support) to the uniform measure of mass 1 on the boundary of the unit disc $$D(0,1)$$.
The purpsoe of this note is to place Bilu’s result in the context of potential theory. The set $$E= D(0,1)$$ has logarithmic capacity $$\gamma(E)= 1$$. We show that for an arbitrary compact set $$E\subset \mathbb{C}$$ with capacity $$\gamma(E)= 1$$, an analogue of Bilu’s theorem is true.
This theorem can undoubtedly be strengthened to allow more general sets $$E_v$$ at non-Archimedean places. It can probably also be extended to adelic sets on curves, with capacities computed relative to several poles [see the author, Capacity theory on algebraic curves, Lect. Notes Math. 1378, Berlin: Springer (1989; Zbl 0679.14012)].
For the entire collection see [Zbl 0922.00026].

##### MSC:
 11G50 Heights 14G40 Arithmetic varieties and schemes; Arakelov theory; heights
##### Keywords:
compact sets of capacity 1; potential theory