Equidistribution in the unit interval of numbers in a real field with bounded Mahler measure

Question

Let $K$ be a real number field, together with a fixed immersion in $\mathbb{R}$, and for each positive real number $M$ consider the set $S_M(K)$ of elements in $K \cap [0,1]$ having Mahler measure smaller than $M$.

When $K = \mathbb{Q}$, the $S_M(K)$ are a Farey sequence, and for $M \rightarrow \infty$ they have been proved to become equidistribuited by Neville ("The Structure of Farey Series", Proceedings of the London Mathematical Society, 1949), with a geometrical method with does not seem to generalize easily. On the other hand, when $K$ is a real quadratic extension of $\mathbb{Q}$ come computational experiment seems to show that the $S_M(K)$ are still uniformly distributed in the unit interval.

Is any there anything known about this problem, or possibly about connected issues, other than Neville's result? Thanks!

Answer 1

This kind of problem has attracted a lot of papers in the last 20 years, albeit in a more geometric framework.

The question, given a projective variety $X$ defined over a number field~$K$, is to understand the number $N(B)$ of rational points of height $\leq B$, when $B\to\infty$, and their distribution in the adelic space $X(A_K)$ of $X$. When $X$ is smooth and Fano, and is anticanonically embedded, a conjecture of Manin predicts the growth of $N(B)$, namely $N(B)\sim c B (\log B)^{t-1}$, where $t\geq 1$ is the rank of the Picard group of $X$ and $c$ is a positive constant. Beside giving an explicit formula for the conjectural constant $c$, Peyre predicts that the $N(B)$ points of height $\leq B$ equidistribute in the adelic space $X(A_K)$ with respect to the Tamagawa measure he has defined.

These conjectures have been proved in many cases, with deep profs (but a counterexample shows that the picture is more complicated).

Anyway, when $X$ is the projective line, the Mahler measure is the height and for that variety, the conjecture is fully proved (and there are several proofs of that case). The limit measure is proportional to ${{\rm d}x\over \max(1,|x|)^2}$ —- if you condition your algebraic numbers to lie in a fixed interval, multiply this measure by the indicator function of that interval.

It implies in particular the result you predicted, but also many others: any number field, and you can independently equidistribute the points at finitely many places (archimedean or not).

As first references, I would sugest you to look at two surveys by Emmanuel Peyre which you can get from Numdam.

Points de hauteur bornée et géométrie des variétés (d'après Y. Manin et al.). Séminaire Bourbaki, 43 (2000-2001), Exposé No. 891, 22 p.

Points de hauteur bornée, topologie adélique et mesures de Tamagawa. Journal de théorie des nombres de Bordeaux, 15 no. 1 (2003), p. 319-349

The second one discusses a little bit equidistribution (Remarque 4.2); the proof in the case you're interested is done in Peyre's paper (Duke Math. J., 1995) using Eisenstein series, and also in my paper with Yuri Tschinkel (Invent. Math., 2002) using additive adelic harmonic analysis.

Answer 2

I think that Farey series were first proved to be uniformly distributed modulo one by Weyl, in his early work on the topic. Maybe his methods apply to the more general situation.