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].

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].

Reviewer: O.Ninnemann (Berlin)