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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics