A question asked by Erdős at the DMV conference in Berlin 1987 (as reported in [Mu11]). Achieving density $1/3$ is trivial, taking $A$ to be all multiples of $3$.

Müller [Mu11] settled this question in the affirmative: in fact one can take $A$ to be the set of all integers congruent to $3\cdot 2^i\pmod{2^{i+2}}$ for any $i\geq 0$, which has density $1/2$. Müller also proved this is best possible, in that $A$ with the property in the question has lower density at most $1/2$.

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This page was last edited 20 January 2026.