On flat trigonometric sums and ergodic flow with simple Lebesgue spectrum

Abstract

A complex polynomial $P(z)=c_0+c_{1}z+...+c_n{z}^n$ is called unimodular if $|c_j|=1$, $j=0,...,n$. Littlewood asked the question (1966) on how close a unimodular polynomial come to satisfying $|P(z)|\approx \sqrt{n+1}$ if $n\ge 1$? In this paper we show that for a given $0<a<b$ and $\eps > 0$ there exist trigonometric sums $\cP(t) = n^{-1/2} \sum_{j=0}^{n-1} \exp(2πi tω(j))$ with a real frequency function $\omega (j)$ which are $\eps$-flat on segment $[a,b]$ acording to the norm in $L^1([a,b])$ (as well as in $L^2([a,b])$). We apply this method to construct a dynamical system having simple spectrum and Lebesgue spectral type in the class of rank-one flows.

19 pages; 2 figures

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