  For a finite set $A\subset \mathbb{R}$ and real $\lambda$, let $A+\lambda
A:=\{a+\lambda b :\, a,b\in A\}$. Combining a structural theorem of Freiman on
sets with small doubling constants together with a discrete analogue of
Pr\'ekopa--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 $\liminf |A+\lambda A|/|A|$ for an
arbitrary algebraic $\lambda$. 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.
