Tight lower bound on $\mathrm{\mid }A+\lambda A\mathrm{\mid }$ for algebraic integer $\lambda$

We prove an asymptotically tight lower bound on $\mathrm{\mid }A+\lambda A\mathrm{\mid }$ for $A\subset \mathbb{C}$ and algebraic integer $\lambda$. The proof combines strong version of Freiman’s theorem, structural theorem on dense subsets of a hypercubic lattice and a generalisation of the continuous result on tight bound for the measure of $K+\tau K$ for a compact subset $K\subset {\mathbb{R}}^d$ of unit Lebesgue measure and a fixed linear operator $\tau \text{\hspace{0.17em}⁣}:{\mathbb{R}}^d\to {\mathbb{R}}^d$, obtained in our previous work.