三角多项式的根与Erdens-Turán定理
Roots of trigonometric polynomials and the Erdős-Turán theorem
摘要：
我们证明，非正式地说，这不是一个巧合，$\cos{(n θ)} + 1 \geq 0$和根的$z^n + 1 =0$是均匀分布的角度-一个版本的声明适用于所有三角多项式与“几个”真正的根源。Erdens-Turán定理指出，如果$p(z) =\sum_{k=0}^{n}{a_k z^k}$被适当地归一化，并且对于$|z|=1$来说不是太大，那么它的根围绕$|z| = 1$聚集，并且在尺度$\sim n^{-1/2}$上在角度上均匀分布。我们建立了一个连接之间的速度均匀分布的根的角度和相应的三角多项式$q(θ) = \Re \sum_{k=0}^{n}{a_k e^{i k θ}}$的符号变化的数量。如果$q(θ)$对于某些$0 < δ< 1/2$具有$\lesssim n^δ$根，则$p(z)$的根在尺度$\sim n^{-(1-δ)} \ll n^{-1/2}$上不经常在角度上聚集。
英文摘要：
We prove, informally put, that it is not a coincidence that $\cos{(n θ)} + 1 \geq 0$ and that the roots of $z^n + 1 =0$ are uniformly distributed in angle -- a version of the statement holds for all trigonometric polynomials with `few' real roots. The Erdős-Turán theorem states that if $p(z) =\sum_{k=0}^{n}{a_k z^k}$ is suitably normalized and not too large for $|z|=1$, then its roots are clustered around $|z| = 1$ and equidistribute in angle at scale $\sim n^{-1/2}$. We establish a connection between the rate of equidistribution of roots in angle and the number of sign changes of the corresponding trigonometric polynomial $q(θ) = \Re \sum_{k=0}^{n}{a_k e^{i k θ}}$. If $q(θ)$ has $\lesssim n^δ$ roots for some $0 < δ< 1/2$, then the roots of $p(z)$ do not frequently cluster in angle at scale $\sim n^{-(1-δ)} \ll n^{-1/2}$.
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