On the ergodicity of Weyl sum cocycles

Gernot Greschonig, Mahesh Nerurkar, Dalibor Volný

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1 Scopus citations

Abstract

We present the quadratic Weyl sums ∑k=0n-1 e 2πi(k2θ+2kx) with θ, x ∈ [0, 1) as cocycles over a measure-preserving transformation on the two-dimensional torus. We show then that these cocycles are not coboundaries for every irrational θ ∈ [0, 1), and that for a dense Gδ set of θ ∈ [0, 1) the corresponding skew product is ergodic. For each of those θ, there exists a dense Gδ set of full measure of x ∈ [0, 1) for which the sequence ∑k=0n-1 e2πi(k2θ+2kx), n = 1, 2, . . . , is dense in ℂ.

Original languageEnglish (US)
Pages (from-to)1851-1863
Number of pages13
JournalErgodic Theory and Dynamical Systems
Volume27
Issue number6
DOIs
StatePublished - Dec 2007

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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