逆和集定理与U^3模逆定理的等价性
An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm
摘要：
我们建立了逆和集定理（可视为近似（阿贝尔）群的分类）和逆定理的Gowers规范（可视为近似多项式的分类）之间的对应关系。特别地，我们证明了Freiman型逆和集定理等价于已知的关于Gowers U^3范数的逆结果，而且前者的多项式加强也等价于后者的多项式加强.我们在两个模型中建立了这个等价，即有限域向量空间F_2 ^n和循环群Z/NZ。在这两种情况下，论证都涉及澄清某些类型的近似同态的结构。
英文摘要：
We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freiman type are equivalent to the known inverse results for the Gowers U^3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces F_2^n, and of the cyclic groups Z/NZ. In both cases the argument involves clarifying the structure of certain types of approximate homomorphism.
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