On the size of A+λA for algebraic λ

For a finite set A ⊂ R and real λ, let A+λA:={a+λb:a, b ∈ A}. We formulate a conjecture about the value of lim inf |A + λA|/|A| for an arbitrary algebraic λ. We support this conjecture by proving a tight lower bound on the Lebesgue measure of K +T K for a given linear operator T ∈ End(R^d) and a compact set K ⊂ R^d with fixed measure. This continuous result also yields an upper bound in the conjecture. Combining a structural theorem of Freiman on sets with small doubling constants together with a novel discrete analogue of the Prékopa–Leindler inequality we prove a lower bound |A +^√2A| ⩾ (1 +^√2)² |A| − O(|A|^1−ε), which is essentially tight. This proves the conjecture for the specific case λ =^√2.