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Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with\[\lim \frac{a_{n+1}}{a_n}=2\]such that\[P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}\]has density $1$ for every cofinite subsequence $A'$ of $A$?
This has been solved in the affirmative by ebarschkis in the comments (based on idea of Tao and van Doorn, also in the comments).
2025-10-20 00:00:00
Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with\[\lim \frac{a_{n+1}}{a_n}=2\]such that\[P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}\]has density $1$ for every cofinite subsequence $A'$ of $A$?
