Summary: For any real number \(\alpha \geq 1\) 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 [Eur. J. Comb. 34, No. 8, 1293–1296 (2013; Zbl 1364.11027)] proved a result on contraction maps and employed it to prove that \(|A+\alpha \cdot A|\geq \frac 1{8} \alpha |A|+o(|A|)\). In this paper, we improve Breuillard and Green’s result on contraction maps and use it to prove that \(|A+\alpha\cdot A|\geq (\alpha +1) |A| +o(|A|)\). The multiplicative constant \(\alpha +1\) is the best possible. We also pose two problems for further research. (11 Refs.)