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

Abstract

By A(xn)={∑n=1∞ɛnxn:ɛn=0,1} we denote the achievement set of the absolutely convergent series ∑n=1∞xn. We study the relation between the achievement set of the series on the plane and the achievement sets of its projection into two coordinates. We mainly focus on the series ∑n=1∞(xn,yn) where (xn) is a geometric series and yn = xσ(n) for some permutation σ ∈ S∞.If (xn) is a multigeometric sequence, then A(xn, xσ(n)) can be one of at least seven types of sets, which are strongly related to three types of attainable achievement sets on the real line. We conjecture that if (xn) multigeometric, then A(xn, xσ(n)) can be one of eight types – none of them homeomorphic to the other one.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∞.