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

TL;DR

This paper provides a straightforward proof of the existence of $L^1$-flat polynomials with coefficients 0 and 1, and constructs examples of conservative ergodic maps with simple Lebesgue spectrum, addressing open questions in harmonic analysis and dynamical systems.

Contribution

It offers a simple proof for the existence of $L^1$-flat polynomials with specific coefficients and constructs ergodic maps with simple Lebesgue spectrum, answering longstanding open questions.

Findings

Existence of $L^1$-flat polynomials with coefficients 0,1 confirmed.

Constructed conservative ergodic maps with simple Lebesgue spectrum.

Answered questions related to Bourgain's problem and Lehmer's problem.

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.