Excess decay for minimizing hypercurrents mod 2Q

We consider codimension 1 area-minimizing m-dimensional currents T mod an even integer p=2Q in a C2 Riemannian submanifold Σ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point q∈spt(T)∖sptp(∂T) where at least one such tangent cone is Q copies of a single plane. While an analogous decay statement was proved in arXiv:2111.11202 as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of Σ. This technical improvement is in fact needed in arXiv:2201.10204 to prove that the singular set of T can be decomposed into a C1,α (m−1)-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most m−2.