关于F2n中的和集

设F₂为两个元素组成的有限域, F₂ⁿ 为F₂上的n维向量空间. 对于集合A, B ⊆ F₂ⁿ , 它们的和集定义为所有两两互异的和a+b所组成的集合, 其中a∈A, b∈B. Green 和Tao 证明了: 设K > 1,如果A, B ? F₂ⁿ 且|A + B|≤K|A|^1/2|B|^1/2, 则存在一个子空间H?F₂ⁿ 满足
|H|>>exp(-O(√KlogK))|A|
以及x,y∈F₂ⁿ, 使得
|A∩(x+H)|^1/2|B∩(y+H)|^1/2≥1/2K|H|.
本文我们将使用Green 和Tao 的方法并作一些修改, 证明如果|H|>>exp(-O(√K))|A|,
则以上的结论仍然成立.

On sumsets in F2n

Let F₂ be the finite field of two elements, F₂ⁿ be the vector space of dimension n over F₂. For sets A, B F₂ⁿ, their sumset is defined as the set of all pairwise sums a + b with a ∈ A, b ∈ B.
Green and Tao proved that, let K > 1, if A, B ⊆ F₂ⁿ and |A + B|≤K|A|^1/2|B|^1/2, then there exists a subspace H ⊆ F₂ⁿ with
|H|>>exp(-O(√KlogK))|A|
and x, y∈F₂ⁿ such that
|A∩(x+H)|^1/2|B∩(y+H)|^1/2≥1/2K|H|.
In this note, we shall use the method of Green and Tao with some modi cation to prove that if
|H|>>exp(-O(√K))|A|,
then the above conclusion still holds true.