On the ergodicity of infinite antisymmetric extensions of symmetric IETs

We consider skew product extensions over symmetric interval exchange transformations on the unit interval [0, 1) with respect to the cocycle f=\chi_{(0,1/2)}-\chi_{(1/2,1)} . We prove that for almost every interval exchange transformation T with symmetric combinatorial data, the skew product T_f\colon [0, 1) \times \Z \to [0, 1) \times \Z given by T_f(x,r)=(T(x),r+f(x)) is ergodic with respect to the product of the Lebesgue and counting measures.