Let $A=\{1\leq a_1 < a_2 < \cdots\}$ and $B=\{1\leq b_1 < b_2 < \cdots\}$ be sets of integers with $a_n/b_n\to 1$.
If $A+B$ contains all sufficiently large positive integers then is it true that $\limsup 1_A\ast 1_B(n)=\infty$?
Formalization note: There's some discussion in the comments of [erdosproblems.com/28] and [erdosproblems.com/1145] about whether or not $0$ should be included in $A$ or $B$ and has been left purposely ambiguous. Problem 1145 was originally written as $A + B = \mathbb{N}$, which would imply that $0$ would need to exist in $A$ or $B$ to include $1$ in $A + B$. However, it's been made more general and rewritten as "sufficiently large positive integers". The formalization below is the version that includes $0$.
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Instances For
Let $A=\{1\leq a_1 < a_2 < \cdots\}$ and $B=\{1\leq b_1 < b_2 < \cdots\}$ be sets of integers with $a_n/b_n\to 1$.
If $A+B$ contains all sufficiently large positive integers then is it true that $\limsup 1_A\ast 1_B(n)=\infty$?
A conjecture of Erdős and Sárközy.
A stronger form of [erdosproblems.com/28].