Erdős Problem #1145
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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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 \(A + B = \mathbb{N}\),
Attempts
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Proved the critical quadratic-window reduction and excluded every fixed-base digit complement; exhaustively computed the sharp balanced perfect-complement tail distortion for all 2<=m<=30, but the uni
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