Problem 254

Erdős Problem #254: declared status 'open'. Formalized: yes. 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 $A$. Prize: no. Tags: number theory.