Problem Statement
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 $A$.
Categories:
Number Theory
Progress
Cassels [Ca60] proved this under the alternative hypotheses\[\lim \frac{\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert}{\log\log x}=\infty\]and\[\sum_{n\in A} \{ \theta n\}^2=\infty\]for every $\theta\in (0,1)$.Source: erdosproblems.com/254 | Last verified: January 14, 2026