All Random Solved Random Open
Let $A\subseteq \mathbb{N}$ be such that \[\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty\] and \[\sum_{n\in A} \{ \theta n\}=\infty\] for every $\theta\in (0,1)$, where $\{x\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.
Cassels [Ca60] proved this under the alternative hypotheses \[\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert\gg \log\log x\] and \[\sum_{n\in A} \{ \theta n\}^2=\infty\] for every $\theta\in (0,1)$.