$L^1$-flat polynomials and simple Lebesgue spectrum for conservative maps exist: A simple proof
Abstract
We present a simple proof on the existence of -flat analytic polynomials with coefficients 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 -norm of such polynomials and to a question inspired by Lehmer's problem on the supremum of the Mahler measures of those polynomials.
Cite
@article{arxiv.2210.15480,
title = {$L^1$-flat polynomials and simple Lebesgue spectrum for conservative maps exist: A simple proof},
author = {el Houcein el Abdalaoui},
journal= {arXiv preprint arXiv:2210.15480},
year = {2023}
}
Comments
A 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