On the two-dimensional Kronecker-sequence and a class of ergodic skew- products (Q1337775)

Transformations \[ T_ \phi: \mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}\times \mathbb{R}\to \mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}\times \mathbb{R} \] of the form \(T_ \phi(x, y,z)= (x+ \alpha, y+ \beta, z+\phi(x, y))\) are considered, where \(\alpha\), \(\beta\) are real numbers such that 1, \(\alpha\), \(\beta\) are linearly independent over the rationals and \(\phi(x, y)= x\sin 2\pi y\). It is proved that for uncountable many \(\alpha\) there exists uncountable many \(\beta\) such that \(T_ \phi\) is ergodic with respect to the standard product measure.