On the density function of the distribution of real algebraic numbers

Koleda, Denis

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 179-200.

Dans cet article, nous étudions la distribution des nombres algébriques réels. Étant donné un intervalle $I$, un entier positif $n$ et $Q>1$, on définit la fonction ${\Phi }_n(Q;I)$ comme étant le nombre de nombres algébriques dans $I$ de degré $n$ et hauteur naïve $\le Q$. Soit $I_x=(-\infty ,x]$. La fonction de distribution est définie comme la limite (quand $Q\to \infty$) de ${\Phi }_n(Q;I_x)$ divisé par le nombre total de nombres algébriques réels de degré $n$ et de hauteur naïve $\le Q$. Nous montrons que la fonction de distribution existe et est continûment différentiable. Nous donnons aussi une formule explicite pour sa dérivée (dénommée la densité de la distribution). Nous établissons une formule asymptotique pour ${\Phi }_n(Q;I)$ avec des estimations supérieure et inférieure pour le terme d’erreur dans cette formule. Il est démontré que ces estimations sont exactes pour $n\ge 3$. Une conséquence du théorème principal est le fait que la distribution des nombres réels algébriques de degré $n\ge 2$ est non uniforme.

In this paper we study the distribution of the real algebraic numbers. Given an interval $I$, a positive integer $n$ and $Q>1$, define the counting function ${\Phi }_n(Q;I)$ to be the number of algebraic numbers in $I$ of degree $n$ and height $\le Q$. Let $I_x=(-\infty ,x]$. The distribution function is defined to be the limit (as $Q\to \infty$) of ${\Phi }_n(Q;I_x)$ divided by the total number of real algebraic numbers of degree $n$ and height $\le Q$. We prove that the distribution function exists and is continuously differentiable. We also give an explicit formula for its derivative (to be referred to as the distribution density) and establish an asymptotic formula for ${\Phi }_n(Q;I)$ with upper and lower estimates for the error term in the asymptotic. These estimates are shown to be exact for $n\ge 3$. One consequence of the main theorem is the fact that the distribution of real algebraic numbers of degree $n\ge 2$ is non-uniform.

Classification : 11N45, 11J83, 11K38
Mots clés : real algebraic numbers, distribution of algebraic numbers, integral polynomials, generalized Farey sequences

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