Achievement sets on the plane-Perturbations of geometric and multigeometric series

We prove a general fact on the Hausdorff dimension of the achievement set in Banach spaces. As a corollary we obtain that if 0 < q ≤ 1/2, dimH(A(qn, qσ(n))) = dimH(A(xn)) = −log 2/log q for some class of regular permutations σ ∈ S∞.