Freiman theorem for integer sets ($|A+A|\le K|A|⟹A\subseteq P,\mathrm{rank}P\le r(K),|P|\le C(K)|A|$)

A finite nonempty set of integers with bounded doubling lies in a generalized arithmetic progression with rank and relative size bounded only by the doubling constant. A proof models a large subset in a dense cyclic set, finds a Bohr set in its fourfold difference set by Fourier analysis, and uses the progression in a Bohr set from successive minima. The progression lifts through a sufficiently high-order Freiman isomorphism. A greedy covering of the original set by translates of its doubled progression then gives the claimed containing progression. Geometry-of-numbers properification may make the final progression proper.