Abstract
A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space
Highlights
The Polynomial Freiman-Ruzsa conjecture, first suggested by Katalin Marton, would, if true, give a polynomial relation between combinatorial and algebraic notions of approximate groups
We restrict our attention to subsets of Euclidean space
Sets of small doubling can be viewed as a combinatorial notion of “approximate
Summary
The Polynomial Freiman-Ruzsa conjecture, first suggested by Katalin Marton, would, if true, give a polynomial relation between combinatorial and algebraic notions of approximate groups. In Rn, a natural example is that of lattice points in a symmetric convex body, as the following standard fact shows. Let D be a maximal subset of L ∩ 2B satisfying that the sets in. Freiman [2] showed that sets of small doubling must be contained in a low-dimensional affine subspace. There is an exponential gap between this bound (which is tight) and the example of lattice points in convex bodies. GAPs have a simpler combinatorial structure than the general case of linear images of lattice points in a convex body. As such, it will be pleasing if Conjecture 1.2 can be simplified as follows
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