#### Vol. 15, No. 5, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Additive properties of numbers with restricted digits

### Han Yu

Vol. 15 (2021), No. 5, 1283–1301
##### 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 $𝜖>0$, as $N\to \infty$,

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

##### Keywords
dynamical system, Diophantine equation, Furstenberg's intersection problem
##### Mathematical Subject Classification
Primary: 11A63, 11K55, 28A80, 28D05, 37C45