Abstract

For a finite set $A\subset ℝ$ and real $\lambda$, let $A+\lambda A:=\{a+\lambda b:a,b\in A\}$. We formulate a conjecture about the value of $\mathrm{liminf}<!--FUNCTION\; APPLICATION-->|A+\lambda A|∕\left|A\right|$ for an arbitrary algebraic $\lambda$. We support this conjecture by proving a tight lower bound on the Lebesgue measure of $K+\mathcal{𝒯}K$ for a given linear operator $\mathcal{𝒯}\in \mathrm{End}<!--FUNCTION\; APPLICATION-->({ℝ}^d)$ and a compact set $K\subset {ℝ}^d$ 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 $|A+\sqrt{2}A|\ge {(1+\sqrt{2})}^2\left|A\right|-O({\left|A\right|}^{1-𝜀})$, which is essentially tight. This proves the conjecture for the specific case $\lambda =\sqrt{2}$.

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