Spectral ergodic Banach problem and flat polynomials

Abstract

We exhibit a sequence of flat polynomials with coefficients $0,1$. We thus get that there exist a sequences of Newman polynomials that are $L^{\alpha }$-flat, $0\le \alpha <2$. This settles an old question of Littlewood. In the opposite direction, we prove that the Newman polynomials are not $L^{\alpha }$-flat, for $\alpha \ge 4$. We further establish that there is a conservative, ergodic, $\sigma$-finite measure preserving transformation with simple Lebesgue spectrum. This answer affirmatively a long-standing problem of Banach from the Scottish book. Consequently, we obtain a positive answer to Mahler's problem in the class of Newman polynomials, and this allows us also to answer a question raised by Bourgain on the supremum of the $L^1$-norm of $L^2$-normalized idempotent polynomials.

arXiv commentIn this revised version, misprints, spellings, punctuation and grammar are corrected. Scientific comments and suggestions are welcome!