ROOTS OF TRIGONOMETRIC POLYNOMIALS AND THE ERDOS-TURaN THEOREM

We prove, informally put, that it is not a coincidence that cos(n theta)+1 >= 0 and the roots of zn+1=0 are uniformly distributed in angle-a version of the statement holds for all trigonometric polynomials with "few" real roots. The Erds-Turan theorem states that if p(z)=Sigma k=0n</mml:msubsup>akzk is suitably normalized and not too large for |z|=1, then its roots are clustered around |z|=1 and equidistribute in angle at scale similar to 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)=R Sigma k=0n<mml:msub>akeik theta. If q(theta) has less than or similar to n delta roots for some 0<<delta><1/2, then the roots of p(z) do not frequently cluster in angle at scale <similar to>n-(1-delta)<< n-1/2.