Sets in ${\mathbb{Z}}^k$ with doubling $2^k+\delta$ are near convex progressions

Peter van Hintum, Hunter Spink, Marius Tiba

For $\delta >0$ sufficiently small and $A\subset {\mathbb{Z}}^k$ with $|A+A|\le (2^k+\delta )|A|$, we show either $A$ is covered by $m_k(\delta )$ parallel hyperplanes, or satisfies $|\widehat{\operat…