We prove, informally put, that it is not a coincidence that cos $(n\theta )+1\ge 0$ and 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 $q(\theta )=\Re {\sum }_{k=0}^n{a}_k{e}^{ik\theta }$. If $q(\theta )$ has $\lesssim n^{\delta }$ roots for some $0<\delta <1/2$, then the roots of $p(z)$ do not frequently cluster in angle at scale $\sim n^{-(1-\delta )}\ll n^{-1/2}$.

中文翻译：

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三角多项式多项式的根和ERDŐS-TURÁN定理

非正式地说，我们证明，cos不是巧合 $（ñ\theta ）+\mathrm{1个}\ge 0$ 和...的根源 ${ž}^{ñ}+\mathrm{1个}=0$以均匀的角度分布-该语句的版本适用于所有实数根为“很少”的三角多项式。Erdős–Turán定理指出，如果$p（ž）={\sum }_{ķ=0}^{ñ}{\mathrm{一种}}_{ķ}{ž}^{ķ}$ 被适当地标准化并且对于 $|ž|=\mathrm{1个}$，然后将其根聚集在周围 $|ž|=\mathrm{1个}$ 并按比例分布角度 $〜{ñ}^{-\mathrm{1个}/2}$。我们在角度的根的均分布率与相应的三角多项式的正负号变化数量之间建立联系$q（\theta ）=\Re {\sum }_{ķ=0}^{ñ}{\mathrm{一种}}_{ķ}{Ë}^{\mathrm{一世}ķ\theta }$。如果$q（\theta ）$ 具有 $\lesssim {ñ}^{\delta }$ 扎根一些 $0<\delta <\mathrm{1个}/2$，然后是 $p（ž）$ 不要经常成比例地成角度聚集 $〜{ñ}^{-（\mathrm{1个}-\delta ）}\ll {ñ}^{-\mathrm{1个}/2}$。