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Additive properties of numbers with restricted digits
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-06-30, DOI: 10.2140/ant.2021.15.1283
Han Yu

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 $𝜖>0$, as $N\to \infty$,

$#\left(\left({B}_{a}+{B}_{b}\right)\cap {B}_{c}\cap \left[1,N\right]\right)=O\left({N}^{𝜖}\right).$