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DISTRIBUTION OF COMPLEX ALGEBRAIC NUMBERS ON THE UNIT CIRCLE F. G¨ otze,∗ A. Gusakova,∗ Z. Kabluchko,† and D. Zaporozhets‡

UDC 519.2

For −π ≤ β1 < β2 ≤ π, denote by Φβ1 ,β2 (Q) the amount of algebraic numbers of degree 2m, elliptic height at most Q, and arguments in [β1 , β2 ], lying on the unit circle. It is proved that Φβ1 ,β2 (Q) = Q

m+1

β2

p(t) dt + O (Qm log Q) ,

Q → ∞,

β1

where p(t) coincides up to a constant factor with density of the roots of a random trigonometrical polynomial. This density is calculated explicitly using the Edelman–Kostlan formula. Bibliography: 15 titles.

1. Introduction The problem of distribution of real and complex algebraic numbers has been studied intensively in the last two decades and turned out to be closely related with distribution of roots of random polynomials. Before describing the problem considered in this paper, we first give a brief review of recent results in this area. Let A denote the field of (all) algebraic numbers over Q, and let An denote the set of algebraic numbers of degree n ∈ N. Note that the set An is countable and any open subset of R or C contains infinitely many algebraic numbers. In order to study the distribution of these algebraic numbers, we need to choose finite (ordered) subsets of An . To this end, we consider a height function h : A → R+ such that for any n ∈ N and Q > 0, there are only finitely many algebraic numbers α of degree n with h(α) ≤ Q. Note that one usually requires (and we always assume) that h(α ) = h(α) for all conjugates of α. A natural question is to determine the asymptotics of the number of all α ∈ An lying in a given subset of R or C and such that h(α) ≤ Q and the degree n is fixed as Q → ∞. This problem has been studied by Masser and Vaaler in [11] with height function being the Mahler measure. Later, Kaliada determined in [9] the asymptotic number of real algebraic numbers ordered by the naive height; the same result for complex algebraic numbers was obtained in [7]. A generalization of the naive height; the weighted lp -norm (which also generalizes the length, the Euclidean norm, and the Bombieri p-norm), was considered in [8]. Another interesting example of heights, the house of algebraic number, was studied in [2], where the distribution of the Perron numbers is considered. In the present note we would like to study the distribution of algebraic numbers on unit circle. Although our methods work for any weighted lp -norm (including the naive height), we consider the weighted Euclidean (or elliptic) height only: this case corresponds to the simplest asymptotic distribution formula. ∗ Bielefeld University, Bielefeld, [email protected]. † ‡

Germany,

e-mail:

[email protected],

M¨ unster University, M¨ unster, Germany, e-mail: [email protected].

St.Petersburg Department [email protected].

of

Steklov

Mathematical

Institute,

St.Petersburg,

Russia,

e-mail:

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 90–107. Original article submitted October 6, 2018. 54