Abstract

We first prove two results which both imply that for any sequence B of asymptotic density zero there exists an infinite sequence A such that the sum of any number of distinct elements of A does not belong to B. Then, for any ε > 0, we construct an infinite sequence of positive integers A = {a_{1} < a_{2} < a_{3} < …} satisfying a_{n} < K(ε)(1 + ε)^{n} for each n ∈ ℕ such that no sum of some distinct elements of A is a perfect square. Finally, given any finite set U ⊂ ℕ, we construct a sequence A of the same growth, namely, a_{n} < K(ε, U)(1 + ε)^{n} for every n ∈ ℕ such that no sum of its distinct elements is equal to uv^{s} with u ∈ U, v ∈ ℕ and s ≥ 2.