Formulations of the PFR Conjecture over $\mathbb{Z}$

The polynomial Fre\u{\i}man--Ruzsa conjecture is a fundamental open question in additive combinatorics. However, over the integers (or more generally $\mathbb{R}^d$ or $\mathbb{Z}^d$) the optimal formulation has not been fully pinned down. The conjecture states that a set of small doubling is controlled by a very structured set, with polynomial dependence of parameters. The ambiguity concerns the class of structured sets needed. A natural formulation in terms of generalized arithmetic progressions was recently disproved by Lovett and Regev. A more permissive alternative is in terms of \emph{convex progressions}; this avoids the obstruction, but uses is a significantly larger class of objects, yielding a weaker statement. Here we give another formulation of PFR in terms of Euclidean ellipsiods (and some variations). We show it is in fact equivalent to the convex progression version; i.e. that the full range of convex progressions is not needed. The key ingredient is a strong result from asymptotic convex geometry.