On the density function of the distribution of real algebraic numbers

Abstract

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 Φ n (Q; I) to be the number of algebraic numbers in I of degree n and height ≤ Q.The distribution function is defined to be the limit (as Q → ∞) of Φ n (Q; I x ) divided by the total number of real algebraic numbers of degree n and height ≤ 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 Φ n (Q; I) with upper and lower estimates for the error term in the asymptotic. These estimates are shown to be exact for n ≥ 3. One consequence of the main theorem is the fact that the distribution of real algebraic numbers of degree n ≥ 2 is non-uniform.

Distribution of real algebraic integers

“…Lemma 5 (see, e.g., [17]). Note that the statement of the lemma implies that Ω n (Q, S) = 0 if S∩(−Q−1, Q+1) = ∅, and Ω n (Q,…”

“…According to [6], the n-th degree Farey sequence of order Q is the sequence of all real roots of the set of integer polynomials of degree (at most) n and height at most Q. The elements of the 1-st degree sequence lying within [0, 1] form the well-known classical Farey sequence (see [17] for details) and tend to be distributed uniformly in [0, 1] as Q → ∞. For n ≥ 2 it turned out [17] that, as Q gets large, the distribution of the n-th degree Farey sequence never tends to be uniform, however, it can be described in terms of a density function.…”

“…The elements of the 1-st degree sequence lying within [0, 1] form the well-known classical Farey sequence (see [17] for details) and tend to be distributed uniformly in [0, 1] as Q → ∞. For n ≥ 2 it turned out [17] that, as Q gets large, the distribution of the n-th degree Farey sequence never tends to be uniform, however, it can be described in terms of a density function. Namely, the following theorem was proved.…”

“…The expression (2) defines a continuous positive piecewise function. For example [15], [17,Remark 2 in Section 4] to represent (2) by an explicit analytic expression for all n: φ n (t) = 2 n−1 1 + 1 3…”

Correlations between real conjugate algebraic numbers

“…In 1971 Brown and Mahler [6] introduced a natural generalization of the Farey sequences: the Farey sequence of degree n and order Q is the set of all real roots of integral polynomials of degree n and height at most Q. The distribution of the generalized Farey sequences has been investigated in [13] (see also [11], [12] for the case n = 2).…”