Some Notes on Flat Polynomials

Abstract

Connection of flat polynomials with some spectral questions in ergodic theory is discussed. A necessary condition for a sequence of polynomials of the type $\frac{1}{\sqrt{N}} \big(1 +\sum_{j=1}^{N-1} z^{n_j}\big)$ to be flat in almost everywhere sense is given, which contrasts with a similar necessary condition for a sequence of polynomials to be ultraflat.