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OPEN
Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\{ \sum\epsilon_k4^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $4$.

Does $A+B$ have positive density?

A problem of Burr, Erdős, Graham, and Li [BEGL96]. More generally, if $n_1<\cdots<n_k$ have \[\sum_{i=1}^k\log_{n_k}(2)>1\] and $A_i$ is the set of integers with only the digits $0,1$ in base $n_i$ then does $A_1+\cdots+A_k$ have positive density?

See also [124].