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arXiv · 1508.06439

Spectral ergodic Banach problem and flat polynomials

el Houcein el Abdalaoui
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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.

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el Houcein el Abdalaoui. 2015-08-26. Spectral ergodic Banach problem and flat polynomials. https://arxiv.org/abs/1508.06439

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