Distribution of Complex Algebraic Numbers on the Unit Circle

For −π ≤ β 1 < β 2 ≤ π, denote by Φ β1,β2 (Q) the amount of algebraic numbers of degree 2m, elliptic height at most Q, and arguments in [β 1 , β 2 ], lying on the unit circle. It is proved that Φ β 1 , β 2 Q = Q m + 1 ∫ β 1 β 2 p t dt + O Q m log Q , Q → ∞ , 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.