欧几里得空间中多项式Freiman-Ruzsa猜想强变式的一个反例
A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space
摘要：
多项式Freiman-Ruzsa猜想是可加组合数学中的中心公开问题之一。如果是真的，它将给出紧密的数量界限有关的组合和代数概念的近似子群。本文主要研究欧几里得空间的子集。在这种情况下，原来的猜想认为近似代数子群作为一个凸体的格点的集合。格林在2007年提出了这样一个问题：是否可以将其简化为一个广义的等差数列，同时在基本参数中不损失超过一个多项式因子。我们给出了一个否定的答案，这个问题的基础上，最近的反向闵可夫斯基定理结合随机格的估计。
英文摘要：
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.
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