  A given subset $A$ of natural numbers is said to be complete if every element
of $\N$ is the sum of distinct terms taken from $A$. This topic is strongly
connected to the knapsack problem which is known to be NP complete. The main
goal of the paper is to study the structure of subset sums in a higher
dimension. We show 'dense' sets and generalized arithmetic progrssions in
subset sums of certain sets.
