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

Abstract

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 $|\widehat{\mathrm{co}}(A)\setminus A|\le c_k\delta |A|$, where $\widehat{\mathrm{co}}(A)$ is the smallest convex progression (convex set intersected with a sublattice) containing $A$. This generalizes the Freiman-Bilu $2^k$ theorem, Freiman's $3|A|-4$ theorem, and recent sharp stability results of the present authors for sumsets in ${\mathbb{R}}^k$ conjectured by Figalli and Jerison.

arXiv comment59 pages, heavily revised, accepted to Advances in Mathematics