A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

TL;DR

This paper provides a counterexample in Euclidean space that disproves a strong version of the Polynomial Freiman-Ruzsa conjecture, showing that approximate algebraic subgroups cannot always be simplified to generalized arithmetic progressions without significant loss.

Contribution

It demonstrates that the strong variant of the Polynomial Freiman-Ruzsa conjecture does not hold in Euclidean space by constructing a counterexample using reverse Minkowski theorems.

Findings

Counterexample disproves the strong conjecture in Euclidean space

Approximate algebraic subgroups cannot always be simplified to generalized arithmetic progressions

Uses reverse Minkowski theorem and lattice estimates for the construction

Abstract

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.