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

TL;DR

This paper explores different formulations of the Polynomial Freman--Ruzsa (PFR) conjecture over integers, establishing an equivalence between convex progression and Euclidean ellipsoid formulations using advanced convex geometry.

Contribution

It introduces a new formulation of the PFR conjecture using Euclidean ellipsoids and proves its equivalence to the convex progression formulation, clarifying the necessary class of structured sets.

Findings

Euclidean ellipsoid formulation is equivalent to convex progression formulation.

The equivalence simplifies the understanding of the PFR conjecture over integers.

Utilizes results from asymptotic convex geometry to establish the equivalence.

Abstract

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.