PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Sums of dilates in the real numbers

Volume 182 / 2018

Yong-Gao Chen, Jin-Hui Fang Acta Arithmetica 182 (2018), 231-241 MSC: Primary 11B13; Secondary 11B75. DOI: 10.4064/aa170221-22-9 Published online: 22 January 2018


For any real number $\alpha \ge 1$ and any finite nonempty subset $A$ of the real numbers, let $\alpha \cdot A=\{ \alpha a \mid a\in A\}$. In 2013, E. Breuillard and B. Green proved a result on contraction maps and employed it to prove that $|A+\alpha \cdot A|\ge \frac 1{8} \alpha |A|+o(|A|)$. In this paper, we improve Breuillard and Green’s result on contraction maps and use it to prove that $|A+\alpha\cdot A|\ge (\alpha +1) |A| +o(|A|)$. The multiplicative constant $\alpha +1$ is the best possible. We also pose two problems for further research.


  • Yong-Gao ChenSchool of Mathematical Sciences and
    Institute of Mathematics
    Nanjing Normal University
    Nanjing 210023, P.R. China
  • Jin-Hui FangDepartment of Mathematics
    Nanjing University of
    Information Science and Technology
    Nanjing 210044, P.R. China

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image