= Freiman theorem for integer sets
{c}
{title2=$|A+A|\le K|A|\implies A\subseteq P,\quad\operatorname{rank}P\le r(K),\quad|P|\le C(K)|A|$}
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.
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