Solution of a problem of P. Turán on zeros of orthogonal polynomials on the unit circle (Q1100602)

It is well known that the zeros of orthogonal polynomials on the unit circle are all inside the unit circle. \textit{P. Turán} [J. Approximation Theory 29, 23-58 (1980; Zbl 0454.41001)] asked whether there is a system of orthogonal polynomials on the unit circle such that the zeros are everywhere dense in \(\{| z| \leq 1\}\). The authors show, using a straightforward method, that there exist positive measures on the unit circle such that the zeros of the corresponding orthogonal polynomials are dense in \(\{| z| \leq 1\}\), thereby giving an affirmative answer to Turán's problem.