Spectral ergodic Banach problem and flat polynomials

TL;DR

This paper constructs flat polynomials with coefficients 0 and 1, resolving longstanding questions about their flatness properties and spectral characteristics, and establishing new links between polynomial flatness and ergodic theory.

Contribution

It demonstrates the existence of flat Newman polynomials in certain $L^eta$ spaces and proves their non-flatness in others, while also constructing a measure-preserving transformation with Lebesgue spectrum.

Findings

Existence of $L^eta$-flat Newman polynomials for $0 extless eta extless 2$

Non-flatness of Newman polynomials for $eta extgreater 4$

Construction of a measure-preserving transformation with Lebesgue spectrum

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, $0 \leq \alpha <2$. This settles an old question of Littlewood. In the opposite direction, we prove that the Newman polynomials are not $L^\alpha$-flat, for $\alpha \geq 4$. 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.