On the size of $A+\lambda A$ for algebraic $\lambda$

Abstract

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 Prékopa--Leindler inequality we prove a lower bound $|A+\sqrt{2}A|\ge (1+\sqrt{2}{)}^2|A|-O({|A|}^{1-\epsilon })$ which is essentially tight. We also formulate a conjecture about the value of $\mathrm{liminf}|A+\lambda A|/|A|$ 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 $\mathcal{T}\in \mathrm{End}({\mathbb{R}}^d)$ 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.

References in corpus (1)

• On the Bogolyubov-Ruzsa lemma