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 $\varepsilon >0,$ we construct an infinite sequence of positive integers $A=\{a_1<a_2<a_3<\dots\}$ satisfying $a_n < K(\varepsilon ) (1+\varepsilon )^n$ for each $n \in \mathbb{N}$ such that no sum of some distinct elements of $A$ is a perfect square. Finally, given any finite set $U \subset \mathbb{N},$ we construct a sequence $A$ of the same growth, namely, $a_n < K(\varepsilon ,U) (1+\varepsilon )^n$ for every $n \in \mathbb{N}$ such that no sum of its distinct elements is equal to $uv^s$ with $u \in U,$ $v \in \mathbb{N}$ and $s \geq 2.$