Erdős Problem #1145

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 $\limsup 1_A\ast 1_B(n)=\infty$?

#1145: [Va99,1.17]

additive combinatorics | additive basis

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A conjecture of Erdős and Sárközy. Some condition relating $A$ and $B$ is necessary since, for example, if $A$ is the set of all integers with only even binary digits and $B$ is the set of all integers with only odd binary digits than $1_A\ast 1_B(n)=1$ for all $n$.

This is a stronger form of [28]. See also [331].

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This page was last edited 08 February 2026.

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Additional thanks to: Felix Pernegger

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