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arXiv · 1811.00996

Distribution of complex algebraic numbers on the unit circle

Friedrich Götze·Anna Gusakova·Zakhar Kabluchko·Dmitry Zaporozhets
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Abstract

For $-π\leqβ_1<β_2\leqπ$ denote by $Φ_{β_1,β_2}(Q)$ the number of algebraic numbers on the unit circle with arguments in $[β_1,β_2]$ 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.

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Friedrich Götze, Anna Gusakova, Zakhar Kabluchko, Dmitry Zaporozhets. 2018-11-02. Distribution of complex algebraic numbers on the unit circle. https://doi.org/10.1007/s10958-020-05064-w

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