  The Polynomial Freiman-Ruzsa conjecture is one of the central open problems
in additive combinatorics. If true, it would give tight quantitative bounds
relating combinatorial and algebraic notions of approximate subgroups. In this
note, we restrict our attention to subsets of Euclidean space. In this regime,
the original conjecture considers approximate algebraic subgroups as the set of
lattice points in a convex body. Green asked in 2007 whether this can be
simplified to a generalized arithmetic progression, while not losing more than
a polynomial factor in the underlying parameters. We give a negative answer to
this question, based on a recent reverse Minkowski theorem combined with
estimates for random lattices.
