Roots of trigonometric polynomials and the Erd\H{o}s-Tur\'an theorem

We prove, informally put, that it is not a coincidence that $\cos{(n \theta)}

1 \geq 0 $a n d t h a t t h e r o o t s o f$ z^n + 1 =0$ are uniformly distributed in angle – a version of the statement holds for all trigonometric polynomials with `few’ real roots. The Erd\H{o}s-Tur'an theorem states that if $p(z) =\sum_{k=0}^{n}{a_k z^k} $i s s u i t a b l y n o r m a l i z e d a n d n o t t o o l a r g e f o r$ |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^{i k \theta}} $. I f$ q(\theta) $h a s$ \lesssim n^{\delta} $r o o t s f o r s o m e$ 0 < \delta < 1/2 $, t h e n t h e r o o t s o f$ p(z)$ do not frequently cluster in angle at scale $\sim n^{- (1 - δ)} ≪ n^{- 1 / 2}$ .

We prove, informally put, that it is not a coincidence that $\cos{(n \theta)}

1 \geq 0 $a n d t h a t t h e r o o t s o f$ z^n + 1 =0$ are uniformly distributed in angle – a version of the statement holds for all trigonometric polynomials with `few’ real roots. The Erd\H{o}s-Tur'an theorem states that if $p(z) =\sum_{k=0}^{n}{a_k z^k} $i s s u i t a b l y n o r m a l i z e d a n d n o t t o o l a r g e f o r$ |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^{i k \theta}} $. I f$ q(\theta) $h a s$ \lesssim n^{\delta} $r o o t s f o r s o m e$ 0 < \delta < 1/2 $, t h e n t h e r o o t s o f$ p(z)$ do not frequently cluster in angle at scale $\sim n^{- (1 - δ)} ≪ n^{- 1 / 2}$ .

文献摘要