论文标题
较高维度的差异集
Difference sets in higher dimensions
论文作者
论文摘要
令$ d \ geq 3 $为自然数字。我们表明，对于所有有限的，非空的设置$ a \ subseteq \ mathbb {r}^d $不包含在超平面的翻译中，我们有\ [| a-a | \ geq（2d-2）| a | -o_d（| a |^{1-δ}），\]，其中$Δ> 0 $仅取决于$ d $。这取决于Freiman，Heppes和Uhrin的早期结果，并朝着Stanchescu的猜想迈进了进步。
Let $d \geq 3$ be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- δ}),\] where $δ>0$ is an absolute constant only depending on $d$. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu.
