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

For $\delta >0$ sufficiently small and $A\subset {\mathbb{Z}}^k$ with $|A+A|\le (2^k+\delta)|A|$,weshoweither$A$iscoveredby$m_k(\delta)$ parallel hyperplanes, or satisfies $|\widehat{\operatorname{co}}(A)\setminus A|\le c_k\delta |A|$,where$\widehat{\operatorname{co}}(A)$ is the smallest convex progression (convex set intersected with a sublattice) containing $A$. This generalizes the Freiman-Bilu $2^k$ theorem, Freiman’s $3\mathrm{\mid }A\mathrm{\mid }-4$ theorem, and recent sharp stability results of the present authors for sumsets in ${\mathbb{R}}^k$ conjectured by Figalli and Jerison.