Strongly complete sets and a conjecture of Erd\H{o}s

A set $A\subseteq\mathbb{N}$ is called $\textit{complete}$ if every sufficiently large integer can be written as a sum of distinct elements of $A$. It is $\textit{strongly complete}$ if it remains complete after one deletes finitely many elements from it. We show that $A\subseteq\mathbb{N}$ is strongly complete whenever \[
\big|A\cap(2^k,2^{k+1}]\big|\ge6 \] for every sufficiently large $k\in\mathbb{N}$, and \[
\sum_{a\in A}\|a\theta\|=\infty,
\quad\forall\theta\in\mathbb{R}\setminus\mathbb{Z}. \] In particular, this resolves a 1961 conjecture of Erdős. The proof builds on previous work of Bergelson and Simmons. Our approach also allows us to establish a more general strong-completeness criterion with suitable ordered blocks in place of dyadic intervals.