We give a simple proof of a classical theorem by A. Máté, P. Nevai, and V. Totik on asymptotic behavior of orthogonal polynomials on the unit circle. It is based on a new real-variable approach involving an entropy estimate for the orthogonality measure. Our second result is an extension of a theorem by G. Freud on averaged convergence of Fourier series. We also discuss some related open problems in the theory of orthogonal polynomials on the unit circle.