Sums of dilates in the real numbers
Volume 182 / 2018
Acta Arithmetica 182 (2018), 231-241
MSC: Primary 11B13; Secondary 11B75.
DOI: 10.4064/aa170221-22-9
Published online: 22 January 2018
Abstract
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.
