Regularity of area minimizing currents mod p

We establish a first general partial regularity theorem for area minimizing currents mod(p), for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current mod(p) cannot be larger than m−1. Additionally, we show that, when p is odd, the interior singular set is (m−1)-rectifiable with locally finite (m−1)-dimensional measure.