Ergodicity of skew-products over typical IETs

We prove ergodicity of a class of infinite measure preserving systems, called skew-products. More precisely, we consider systems of the form $$T_f:[0,1)\times \mathbb{R}\to [0,1)\times \mathbb{R},T_f(x,t):=(T(x),t+f(x)),$$ where $T$ is an interval exchange transformation and $f$ is a piece-wise constant function with a finite number of discontinuities. We show that such system is ergodic with respect to ${Leb}_{[0,1)\times \mathbb{R}}$ for a typical choice of parameters of $T$ and $f$.