All sums of hh distinct terms of a sequence

For A⊆ZA⊆Z, we study the gaps in the sequence of all sums of hh pairwise distinct elements of AA. For example, the following result is proved: for any integer h≥3h≥3, there exists A⊆ZA⊆Z such that every integer can be uniquely (neglecting the order) represented as a sum of hh not necessarily distinct elements of AA, and for any integer ℓ≥1ℓ≥1, in the sequence of all sums of ℓℓ pairwise distinct elements of AA, the gaps can be arbitrarily large. Several questions are posed in this paper.