论文标题
$ \ Mathbb {z}^k $带$ 2^k+δ$的集合在凸进度附近
Sets in $\mathbb{Z}^k$ with doubling $2^k+δ$ are near convex progressions
论文作者
论文摘要
对于$δ> 0 $，$ a \ subset \ mathbb {z}^k $带有$ | a+a+a | \ le（2^k+δ）| a | $，我们显示$ a $ a $由$ m_k（δ）$ paralalele hypranes，或满意$ | c_kδ| a | $，其中$ \ wideHat {\ operatorname {co}}（a）$是最小的凸进progression（凸面集与sublattice相交），其中包含$ a $。这概括了Freiman-Bilu $ 2^k $定理，Freiman的$ 3 | A | -4 $定理，以及$ \ Mathbb {r}^k $ Cunienditor in Figalli and Jerison和Jerison conuentions in $ \ Mathbb {r}^k $最近的敏锐稳定性结果。
For $δ>0$ sufficiently small and $A\subset \mathbb{Z}^k$ with $|A+A|\le (2^k+δ)|A|$, we show either $A$ is covered by $m_k(δ)$ parallel hyperplanes, or satisfies $|\widehat{\operatorname{co}}(A)\setminus A|\le c_kδ|A|$, where $\widehat{\operatorname{co}}(A)$ is the smallest convex progression (convex set intersected with a sublattice) containing $A$. This generalizes the Freiman-Bilu $2^k$ theorem, Freiman's $3|A|-4$ theorem, and recent sharp stability results of the present authors for sumsets in $\mathbb{R}^k$ conjectured by Figalli and Jerison.
