Problem 125

Erdős Problem #125: declared status 'disproved'. Formalized: yes. 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? Prize: no. OEIS: A367090. Tags: base representations, number theory.