Strongly complete sets and a conjecture of Erdős

A set $A\subseteq \mathbb{N}$ is called $\mathit{\text{complete}}$ if every sufficiently large integer can be written as a sum of distinct elements of $A$. It is $\mathit{\text{strongly}}\mathit{\text{complete}}$ if it remains complete after one deletes finitely many elements from it. We show that $A\subseteq \mathbb{N}$ is strongly complete whenever $$\mid A\cap (2^k,2^{k+1}]\mid \ge 6$$ for every sufficiently large $k\in \mathbb{N}$, and $$\sum _{a\in A}\mathrm{\parallel }a\theta \mathrm{\parallel }=\mathrm{\infty },\mathrm{\forall }\theta \in \mathbb{R}\setminus \mathbb{Z}\mathrm{.}$$ 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.