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Tight lower bound on $|A+λA|$ for algebraic integer $λ$ (2311.09399v1)

Published 15 Nov 2023 in math.CO

Abstract: We prove an asymptotically tight lower bound on $|A+\lambda A|$ for $A\subset \mathbb{C}$ and algebraic integer $\lambda$. The proof combines strong version of Freiman's theorem, structural theorem on dense subsets of a hypercubic lattice and a generalisation of the continuous result on tight bound for the measure of $K+\tau K$ for a compact subset $K\subset \mathbb{R}d$ of unit Lebesgue measure and a fixed linear operator $\tau\colon\mathbb{R}d\to \mathbb{R}d$, obtained in our previous work.

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