For a finite set $A\subset \mathbb{R}$ and real $λ$, let $A+λA:=\{a+λb :\, a,b\in A\}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of Prékopa--Leindler inequality we prove a lower bound $|A+\sqrt{2} A|\geq (1+\sqrt{2})^2|A|-O({|A|}^{1-\varepsilon})$ which is essentially tight. We also formulate a conjecture about the value of $\liminf |A+λA|/|A|$ for an arbitrary algebraic $λ$. Finally, we prove a tight lower bound on the Lebesgue measure of $K+\mathcal{T} K$ for a given linear operator $\mathcal{T}\in \operatorname{End}(\mathbb{R}^d)$ and a compact set $K\subset \mathbb{R}^d$ with fixed measure. This continuous result supports the conjecture and yields an upper bound in it.