Let $\lambda_1,\ldots,\lambda_k$ be algebraic numbers. We show that
$$|A+\lambda_1\cdot A+\dots+\lambda_k\cdot A|\geq H(\lambda_1,\ldots,\lambda_k)|A|-o(|A|)$$
for all finite subsets $A$ of $\mathbb{C}$, where $H(\lambda_1,\ldots,\lambda_k)$ is an explicit constant that is best possible. In this talk, we will discuss the main tools used in the proof, which include a Frieman-type structure theorem for sets with small sums of dilates, and a high-dimensional notion of density which we call "lattice density". Joint work with David Conlon.
