Distribution of complex algebraic numbers on the unit circle

For $-\pi \le {\beta }_1<{\beta }_2\le \pi$ denote by ${\mathrm{\Phi }}_{{\beta }_1,{\beta }_2}(Q)$ 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 $${\mathrm{\Phi }}_{{\beta }_1,{\beta }_2}(Q)=Q^{m+1}\underset{{\beta }_1}{\overset{{\beta }_2}{\int }}p(t)\mathrm{d}t+O(Q^m\log Q),Q\to \mathrm{\infty },$$ 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.