Additive properties of numbers with restricted digits

Abstract

We consider some additive properties of integers with restricted digit expansions. Let $b\ge 3$ be an integer and $B_b$ be the set of integers whose base $b$ expansions have only digits $\left\{0,1\right\}$. Let $a,b,c$ be three integers greater than $2$. We give some estimates on the size of $\left(B_a+B_b\right)\cap B_c$. In particular, under mild conditions, $\left(B_a+B_b\right)\cap B_c$ is a very thin set in the sense that for each , as $N\to \infty$,

$$\#\left(\left(B_a+B_b\right)\cap B_c\cap \left[1,N\right]\right)=O\left(N^{𝜖}\right).$$