Title: On entropy Marton-type inequalities and small symmetric differences with cosets of abelian groups Show Chinese title

Title: 关于熵马尔顿型不等式与阿贝尔群陪集的小对称差

Abstract: We recognise that an entropy inequality akin to the main intermediate goal of recent works (Gowers, Green, Manners, Tao [3],[2]) regarding a conjecture of Marton provides a black box from which we can also through a short deduction recover another description: if a finite subset $A$ of an abelian group $G$ is such that the distribution of the sums $a+b$ with $(a,b) \in A \times A$ is only slightly more spread out than the uniform distribution on $A$, then $A$ has small symmetric difference with some finite coset of $G$. The resulting bounds are necessarily sharp up to a logarithmic factor. Show Chinese abstract

Abstract: 我们认识到，一种类似于最近工作（Gowers, Green, Manners, Tao [3],[2]）的主要中间目标的熵不等式，关于Marton的一个猜想，提供了一个黑箱，通过一个简短的推导，我们也可以得到另一种描述：如果阿贝尔群$G$的一个有限子集$A$满足和$a+b$的分布与$(a,b) \in A \times A$相比仅稍微比$A$上的均匀分布更分散，那么$A$与$G$的某个有限陪集具有小的对称差。由此得到的界限必然在对数因子范围内是尖锐的。