Problem Statement
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?
Does $A+B$ have positive density?
Categories:
Number Theory Base Representations
Progress
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? Melfi [Me01] noted this is false as written, with a counterexample given by $\{3,9,81\}$, but suggests it is true if we further insist that the $n_k$ are pairwise coprime.If $C=A+B$ then Melfi [Me01] showed $\lvert C\cap[1,x]\rvert \gg x^{0.965}$ and Hasler and Melfi [HaMe24] improved this to $\lvert C\cap [1,x]\rvert \gg x^{0.9777}$. Hasler and Melfi also show that the lower density of $C$ is at most\[\frac{1015}{1458}\approx 0.69616.\]See also [124].
Source: erdosproblems.com/125 | Last verified: January 13, 2026