On flat trigonometric sums and ergodic flow with simple Lebesgue spectrum

2010/02/15 by A. A. Prikhod’ko, Prikhod'ko, A. A. · 2 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #26C05 #28D10 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1002.2808

openalex publication_date 2010/02/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

A complex polynomial P(z) = c₀ + c₁ z +...+ c_n zⁿ 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)| ≈ √(n+1) if n ≥ 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 ∑_j=0^n-1 exp(2πi tω(j)) with a real frequency function ω(j) which are \eps-flat on segment [a,b] acording to the norm in L¹([a,b]) (as well as in L²([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.

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