几何提升与紧致连通阿贝尔群中Freiman的$3k-4$定理
Geometric lifting and Freiman's $3k-4$ theorem in compact connected abelian groups
- The Ohio State University（俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出几何提升方法，在紧致连通阿贝尔群中证明Freiman $3k-4$定理的类比，并建立最优投影定理，解决Christ-Iliopoulou问题，推广至流行和集与Tao卷积不等式。
AI中文摘要：
我们开发了一种几何提升方法，用于解决紧致连通阿贝尔群中的逆和集问题。第一个结果是Freiman的$3k-4$定理的类比，即每个具有足够小Haar测度且满足$\mu_G(A+A)<3\mu_G(A)$的紧致集合$A\subseteq G$都包含在一个一维Bohr集中，该Bohr集的测度至多为$\mu_G(A+A)-\mu_G(A)$。这解决了Christ和Iliopoulou提出的一个问题。证明结合了Bilu定理与溢出论证的几何细化。我们还建立了一个尖锐的投影定理。在具有连通核的连续满射同态下，一个具有足够小正测度且倍增常数至多为$K$（其中$2\le K<3$）的紧致集合，其像的倍增常数至多为$2K-2$，且这个因子是最优的。进一步的结果包括流行和集的$3k-4$定理的变体以及Tao卷积不等式的一个逆定理。
英文摘要：
We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman's $3k-4$ theorem, that is every compact set $A\subseteq G$ of sufficiently small Haar measure satisfying $μ_G(A+A)<3μ_G(A)$ is contained in a one dimensional Bohr set of measure at most $μ_G(A+A)-μ_G(A)$. This resolves a question of Christ and Iliopoulou. The proof combines Bilu's theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most $K$, where $2\le K<3$, has image of doubling at most $2K-2$, and this factor is best possible. Further consequences include variants of the $3k-4$ theorem for popular sumsets and an inverse theorem for Tao's convolution inequality.
