Skew products over translations on ${\mathbf T}^d$, $d \geq 2$

We give an example on ${\mathbf T}^4$ of a minimal translation $R_\alpha$ and a real analytic function $\varphi$, such that the circle-valued skew product extension of $R_\alpha$ by $\varphi$ has a Lebesgue spectrum in the orthocomplement of the space of eigenfunctions.