Distribution of complex algebraic numbers on the unit circle

Abstract

For $-\pi \le {\beta }_1<{\beta }_2\le \pi$ denote by ${\Phi }_{{\beta }_1,{\beta }_2}(Q)$ the number of algebraic numbers on the unit circle with arguments in $[{\beta }_1,{\beta }_2]$ of degree $2m$ and with elliptic height at most $Q$. We show that \[ Φ_{β_1,β_2}(Q)=Q^{m+1}\int\limits_{β_1}^{β_2}{p(t)}\,{\rm d}t+O\left(Q^m\,\log Q\right),\quad Q\to\infty, \] where $p(t)$ coincides up to a constant factor with the density of the roots of some random trigonometric polynomial. This density is calculated explicitly using the Edelman--Kostlan formula.