$L^1$-flat polynomials and simple Lebesgue spectrum for conservative maps exist: A simple proof

Abstract

We present a simple proof on the existence of $L^1$-flat analytic polynomials with coefficients $0,1$ on the circle and on the real line and we give an example of a conservative ergodic map and flow whose unitary operators admits a simple Lebesgue spectrum. Among other results, we obtain an answer to Bourgain's question on the supremum of $L^1$-norm of such polynomials and to a question inspired by Lehmer's problem on the supremum of the Mahler measures of those polynomials.

arXiv commentA simple proof of the positive answer to the old problem asked by Banach from Scottish book is given. In this version, some misprints are corrected and by a counting argument, we get much simpler proof. Scientific comments are welcome! arXiv admin note: text overlap with arXiv:1508.06439