Achievement Sets of Sequences

Given a real sequence (xn), we examine the set of all sums of the form P i∈I xi, as I varies over subsets of the natural numbers. We call this the achievement set of (xn), and write it AS(xn). For instance, AS(1/2) = [0, 1] by the existence of binary expansions, and AS(2/3) is the Cantor middle third set. We explore the properties of these two sequences that account for their very different achievement sets. We give a sufficient condition for a sequence to have an achievement set that is an interval, and another sufficient condition for the achievement set to be a Cantor set. We also examine what sets can occur as achievement sets, and give results on the topology of achievement sets.