An arithmetical property of Rademacher sums

Let ε = {ε_i, i ≥ 1} be a Rademacher sequence, with partial sums S_n = ε₁ +… + ε_n, n ≥ 1. Let N_k be the k-th even integer such that N_k∥S_Nk². We prove that there exists a positive real s, of which the value is explicitly given, such that for any , almost surely.